{
 "license": "CC BY 4.0 — https://creativecommons.org/licenses/by/4.0/",
 "attribution": "Wheel Cue (wheelcue.com)",
 "source": "https://wheelcue.com/research/secret-santa-draw-odds/",
 "slug": "secret-santa-draw-odds",
 "generatedAt": "2026-10-10T14:16:41.001Z",
 "node": "v24.21.0",
 "randomJsSha256": "946e82a3c426d698779d3246c37c732f771db3873c93239ef3952aea39be1e2d",
 "totalSpins": 97173556,
 "rawDraws": 97173556,
 "wheelSpins": 76171953,
 "shuffles": 3017159,
 "seconds": 128.3,
 "checksTotal": 77,
 "checksPassed": 77,
 "checksSimulatedVsExact": 64,
 "checksChiSquare": 10,
 "checksSanity": 3,
 "params": {
  "people": "Person i (0-based in the code, 1-based in the text) is also the i-th to draw. Names are distinct.",
  "hat": {
   "sizes": [
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20
   ],
   "sessions": 200000,
   "sessionsAt10": 1000000,
   "method": "a model of a paper hat drawn in turn: each person in turn spins the name picker over the names still on it and takes the result (Remove & spin); the last person gets the last name without a spin. Same chances as a well-mixed paper hat, but not secret on the wheel: every drawn name shows on screen and in the result history"
  },
  "redraw": {
   "rule": "if anyone draws their own name, the whole hat is drawn again, until nobody has their own name",
   "n": 10,
   "sessions": 150000,
   "uniformityN": 5,
   "uniformitySessions": 200000,
   "triesShown": [
    1,
    2,
    3,
    5
   ]
  },
  "putBack": {
   "rule": "a person who draws their own name puts it back and draws again (Spin again); if the last person is left with only their own name the draw dead-ends and starts over from the top",
   "exactSizes": [
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20
   ],
   "enumeratedSizes": [
    4,
    5,
    6,
    7,
    8,
    9,
    10
   ],
   "chainSizes": [
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20
   ],
   "simulatedSizes": [
    4,
    5,
    6,
    8,
    12,
    15,
    20
   ],
   "sessions": 200000,
   "n": 10,
   "sessionsAtN": 1000000,
   "uniformityN": 5,
   "uniformitySessions": 300000
  },
  "loop": {
   "rule": "press Shuffle once; each person gives to the name below theirs and the last name to the first",
   "n": 10,
   "shuffles": 1000000,
   "uniformityN": 5,
   "uniformityShuffles": 240000
  },
  "couples": {
   "rule": "k couples (persons 1&2, 3&4, ...) who must not draw each other, on top of nobody drawing themselves",
   "tableSizes": [
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20
   ],
   "n": 10,
   "k": 3,
   "helper": {
    "n": 10,
    "k": 3,
    "sessions": 100000,
    "uniformityN": 6,
    "uniformityK": 2,
    "uniformitySessions": 200000
   }
  },
  "lastYear": {
   "rule": "last year's draw was one loop (person 1 gave to 2, 2 to 3, ..., the last to 1); nobody may draw themselves or the person they had last year",
   "tableSizes": [
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11,
    12,
    13,
    14,
    15,
    16,
    17,
    18,
    19,
    20
   ],
   "loopExactSizes": [
    4,
    5,
    6,
    7,
    8,
    9,
    10,
    11
   ],
   "n": 10
  },
  "widgetTableSizes": [
   3,
   4,
   5,
   6,
   7,
   8,
   9,
   10,
   11,
   12,
   13,
   14,
   15,
   16,
   17,
   18,
   19,
   20,
   21,
   22,
   23,
   24,
   25,
   26,
   27,
   28,
   29,
   30,
   31,
   32,
   33,
   34,
   35,
   36,
   37,
   38,
   39,
   40
  ],
  "bruteForceUpTo": 8
 },
 "notes": {
  "simulatedVsExact": "results holds simulated draws (never site usage); theory holds exact math (BigInt counts and exact fractions, rounded to 6 decimals) and is identical on every run. Quote theory values; call results values 'simulated'. Only the fields listed in a row's simulationOnly array have no exact counterpart and are not checked.",
  "siteBehavior": "Wheel Cue has no dedicated Secret Santa tool. The random name picker (/tools/random-name-picker/) has a Shuffle button under the Names box (also Sort A–Z, Reset, Clear); Shuffle reorders the non-empty lines with CueRandom.shuffle (Fisher–Yates), so every order is equally likely. Its option 'Remove the winner after each spin' is OFF by default on that page. The result card offers 'Spin again' (the name stays on the wheel) and 'Remove & spin' (the name comes off and the wheel spins again). A wheel needs at least two entries to spin and holds at most 500 entries; two people with the same name must be told apart (e.g. 'Sam R.' and 'Sam T.') or they are two identical lines. The hat and put-back procedures are modeled on the wheel because it gives exactly the chances of a well-mixed paper hat; the wheel is not a way to run a secret draw in front of the group: every spin shows the drawn name on screen and the result history lists every result. Only the loop (Shuffle) or the organizer's helper, run by one organizer alone, keeps the pairings from the participants, and that organizer sees every pairing. Names typed into a wheel are not stored by the site.",
  "people": "People are numbered in the order they draw (person 1 draws first). Couples are persons 1&2, 3&4, 5&6; last year's draw is the loop 1->2->...->n->1.",
  "spins": "totalSpins counts CueRandom draws that pick a result: one per spin of the wheel (wheelSpins) and n - 1 per Shuffle of n names (shuffles = presses of Shuffle simulated). rawDraws also counts the few draws rejected to remove modulo bias. In the article call totalSpins 'random draws' (drawsLabel), not spins.",
  "checks": "Each check: z = (simulated - exact)/se, pass when |z| <= 4. Chi-square checks ('uniformity' against equal chances, 'goodness of fit' against the exact unequal chances of the put-back rule) list the statistic as simulated, its expected value df as exact and sqrt(2 df) as se, but their z (zMethod 'Wilson–Hilferty') is ((chi/df)^(1/3) - (1 - 2/(9 df))) / sqrt(2/(9 df)), an approximately standard normal score that allows for the skew of a chi-square with few degrees of freedom. With about 80 checks at 4 standard errors, a rerun has roughly a 1 in 200 chance of one check failing by luck alone. Sanity checks (se 0) must hold exactly (the loop never gives anyone themselves or a mutual pair). Means use the exact SD where known (geometric number of tries; self-draws SD = 1), otherwise the sample SD.",
  "rounding": "display strings: two decimals below 10% or above 90%, one decimal between; never 100% or 0% unless exact. Ratios are to 4 decimals."
 },
 "results": {
  "hat": {
   "note": "Simulated hats drawn on the name picker, one spin per person (the last person takes the last name). pNobodySelf = share of hats in which nobody drew their own name. at10 adds the couples rule (persons 1&2, 3&4, 5&6) and last year's loop, counted on the same hats.",
   "rows": [
    {
     "people": 4,
     "sessions": 200000,
     "pNobodySelf": 0.3746,
     "pSomeoneSelf": 0.6254,
     "meanSelf": 0.999865,
     "pExactlyOneSelf": 0.332725
    },
    {
     "people": 5,
     "sessions": 200000,
     "pNobodySelf": 0.36677,
     "pSomeoneSelf": 0.63323,
     "meanSelf": 0.998565,
     "pExactlyOneSelf": 0.37551
    },
    {
     "people": 6,
     "sessions": 200000,
     "pNobodySelf": 0.36667,
     "pSomeoneSelf": 0.63333,
     "meanSelf": 1.00138,
     "pExactlyOneSelf": 0.36728
    },
    {
     "people": 7,
     "sessions": 200000,
     "pNobodySelf": 0.36729,
     "pSomeoneSelf": 0.63271,
     "meanSelf": 0.99897,
     "pExactlyOneSelf": 0.36983
    },
    {
     "people": 8,
     "sessions": 200000,
     "pNobodySelf": 0.368835,
     "pSomeoneSelf": 0.631165,
     "meanSelf": 1.000355,
     "pExactlyOneSelf": 0.36691
    },
    {
     "people": 9,
     "sessions": 200000,
     "pNobodySelf": 0.367215,
     "pSomeoneSelf": 0.632785,
     "meanSelf": 1.00076,
     "pExactlyOneSelf": 0.36807
    },
    {
     "people": 10,
     "sessions": 1000000,
     "pNobodySelf": 0.367474,
     "pSomeoneSelf": 0.632526,
     "meanSelf": 1.000406,
     "pExactlyOneSelf": 0.368257
    },
    {
     "people": 11,
     "sessions": 200000,
     "pNobodySelf": 0.366895,
     "pSomeoneSelf": 0.633105,
     "meanSelf": 1.001195,
     "pExactlyOneSelf": 0.369255
    },
    {
     "people": 12,
     "sessions": 200000,
     "pNobodySelf": 0.36477,
     "pSomeoneSelf": 0.63523,
     "meanSelf": 1.00449,
     "pExactlyOneSelf": 0.36982
    },
    {
     "people": 13,
     "sessions": 200000,
     "pNobodySelf": 0.366195,
     "pSomeoneSelf": 0.633805,
     "meanSelf": 1.002805,
     "pExactlyOneSelf": 0.36887
    },
    {
     "people": 14,
     "sessions": 200000,
     "pNobodySelf": 0.36754,
     "pSomeoneSelf": 0.63246,
     "meanSelf": 0.999435,
     "pExactlyOneSelf": 0.369145
    },
    {
     "people": 15,
     "sessions": 200000,
     "pNobodySelf": 0.37037,
     "pSomeoneSelf": 0.62963,
     "meanSelf": 0.9976,
     "pExactlyOneSelf": 0.365795
    },
    {
     "people": 16,
     "sessions": 200000,
     "pNobodySelf": 0.36895,
     "pSomeoneSelf": 0.63105,
     "meanSelf": 0.99719,
     "pExactlyOneSelf": 0.367285
    },
    {
     "people": 17,
     "sessions": 200000,
     "pNobodySelf": 0.36792,
     "pSomeoneSelf": 0.63208,
     "meanSelf": 1.0002,
     "pExactlyOneSelf": 0.36759
    },
    {
     "people": 18,
     "sessions": 200000,
     "pNobodySelf": 0.367095,
     "pSomeoneSelf": 0.632905,
     "meanSelf": 1.00306,
     "pExactlyOneSelf": 0.36676
    },
    {
     "people": 19,
     "sessions": 200000,
     "pNobodySelf": 0.369835,
     "pSomeoneSelf": 0.630165,
     "meanSelf": 0.99518,
     "pExactlyOneSelf": 0.3672
    },
    {
     "people": 20,
     "sessions": 200000,
     "pNobodySelf": 0.367865,
     "pSomeoneSelf": 0.632135,
     "meanSelf": 0.99934,
     "pExactlyOneSelf": 0.36791
    }
   ],
   "at10": {
    "sessions": 1000000,
    "couples": {
     "k": 3,
     "pValid": 0.184711,
     "pNoSelfButCoupleBreaks": 0.182763,
     "pCoupleBreaksGivenNoSelf": 0.497349
    },
    "lastYear": {
     "pValid": 0.121117,
     "pLastYearBreaksGivenNoSelf": 0.670407
    }
   }
  },
  "redraw": {
   "n": 10,
   "sessions": 150000,
   "meanTries": 2.7189,
   "pMoreThan3Tries": 0.251647,
   "pFirstGivesToLast": 0.111127,
   "receiverOfFirst": [
    0.111027,
    0.111567,
    0.11098,
    0.11124,
    0.111107,
    0.11182,
    0.110207,
    0.110927,
    0.111127
   ],
   "pMutualPair": 0.392913,
   "pSingleLoop": 0.271467,
   "meanLoops": 2.024807,
   "uniformity": {
    "people": 5,
    "sessions": 200000,
    "validDraws": 44,
    "chiSquare": 42.8,
    "df": 43,
    "fewest": 4387,
    "most": 4691
   },
   "note": "Each session redraws the whole hat until nobody has their own name. receiverOfFirst[j] = share of sessions in which person 1 gave to person j+2. fewest/most are this run's counts of the least and most frequent of the valid draws and change on every run."
  },
  "putBackUniformity": {
   "people": 5,
   "attempts": 300000,
   "completed": 260624,
   "validDraws": 44,
   "chiSquareVsExact": 28.3,
   "chiSquareVsEqual": 26917.6,
   "df": 43,
   "expectedChiSquareVsEqualApprox": 27206,
   "note": "All three numbers here are SIMULATED and change on every run. chiSquareVsExact tests the simulated draws against the exact put-back chances (should be near df). chiSquareVsEqual is the simulated chi-square against 'every valid draw equally likely' and is far above df: the put-back rule is not fair. expectedChiSquareVsEqualApprox = df + completed × k × sum (p - 1/k)^2 is only a large-sample approximation of what chiSquareVsEqual should be near; do not quote it as the measured statistic. For the size of the bias quote the exact, deterministic values in theory.putBack.biasRows (totalVariation, wholeDrawMostOverLeast, lastGivesToPreviousRatio)."
  },
  "putBack": {
   "note": "Simulated put-back draws on the name picker. People draw in order 1..n. completed = attempts that did not dead-end; the shares about who gets whom are among completed draws. Every row's dead-end share is checked against the exact DP. From 10 people up, the last person's row, person 1 to person n and the last two swapping are checked against exact values (enumeration at 10, the exact chain above 10); only the fields listed in a row's simulationOnly (mutual pair, single loop, loops above 10 people) have no exact value here and are not checked. lastRow[j] = share in which the last person got person j+1.",
   "rows": [
    {
     "people": 4,
     "attempts": 200000,
     "completed": 172449,
     "pDeadEnd": 0.137755,
     "meanPutBacksPerAttempt": 1.06121
    },
    {
     "people": 5,
     "attempts": 200000,
     "completed": 173442,
     "pDeadEnd": 0.13279,
     "meanPutBacksPerAttempt": 1.068145
    },
    {
     "people": 6,
     "attempts": 200000,
     "completed": 177525,
     "pDeadEnd": 0.112375,
     "meanPutBacksPerAttempt": 1.081315
    },
    {
     "people": 8,
     "attempts": 200000,
     "completed": 182086,
     "pDeadEnd": 0.08957,
     "meanPutBacksPerAttempt": 1.08526
    },
    {
     "people": 10,
     "attempts": 1000000,
     "completed": 924458,
     "pDeadEnd": 0.075542,
     "meanPutBacksPerAttempt": 1.087971,
     "lastGivesToPrevious": 0.179327,
     "lastGivesToFirst": 0.09192,
     "firstGivesToLast": 0.119823,
     "lastTwoSwap": 0.015684,
     "pMutualPair": 0.392158,
     "pSingleLoop": 0.27202,
     "meanLoops": 2.022356,
     "lastRow": [
      0.09192,
      0.092849,
      0.094736,
      0.096873,
      0.099765,
      0.104506,
      0.111379,
      0.128646,
      0.179327
     ],
     "fair": 0.111111
    },
    {
     "people": 12,
     "attempts": 200000,
     "completed": 187139,
     "pDeadEnd": 0.064305,
     "meanPutBacksPerAttempt": 1.087685,
     "lastGivesToPrevious": 0.149766,
     "lastGivesToFirst": 0.076489,
     "firstGivesToLast": 0.098665,
     "lastTwoSwap": 0.010554,
     "pMutualPair": 0.391896,
     "pSingleLoop": 0.228055,
     "meanLoops": 2.178001,
     "lastRow": [
      0.076489,
      0.077397,
      0.077568,
      0.078621,
      0.080945,
      0.082313,
      0.084691,
      0.088768,
      0.095282,
      0.10816,
      0.149766
     ],
     "fair": 0.090909,
     "simulationOnly": [
      "pMutualPair",
      "pSingleLoop",
      "meanLoops"
     ]
    },
    {
     "people": 15,
     "attempts": 200000,
     "completed": 189278,
     "pDeadEnd": 0.05361,
     "meanPutBacksPerAttempt": 1.087645,
     "lastGivesToPrevious": 0.122011,
     "lastGivesToFirst": 0.061259,
     "firstGivesToLast": 0.076205,
     "lastTwoSwap": 0.007407,
     "pMutualPair": 0.391498,
     "pSingleLoop": 0.181241,
     "meanLoops": 2.379907,
     "lastRow": [
      0.061259,
      0.061143,
      0.062226,
      0.062332,
      0.063325,
      0.062828,
      0.064218,
      0.066542,
      0.066542,
      0.068745,
      0.071905,
      0.0776,
      0.089324,
      0.122011
     ],
     "fair": 0.071429,
     "simulationOnly": [
      "pMutualPair",
      "pSingleLoop",
      "meanLoops"
     ]
    },
    {
     "people": 20,
     "attempts": 200000,
     "completed": 191621,
     "pDeadEnd": 0.041895,
     "meanPutBacksPerAttempt": 1.08194,
     "lastGivesToPrevious": 0.092547,
     "lastGivesToFirst": 0.046613,
     "firstGivesToLast": 0.0549,
     "lastTwoSwap": 0.003924,
     "pMutualPair": 0.391758,
     "pSingleLoop": 0.13558,
     "meanLoops": 2.642706,
     "lastRow": [
      0.046613,
      0.046644,
      0.046989,
      0.046075,
      0.045778,
      0.046467,
      0.047803,
      0.047599,
      0.047463,
      0.048601,
      0.048601,
      0.050057,
      0.050391,
      0.051659,
      0.053987,
      0.055698,
      0.059628,
      0.067399,
      0.092547
     ],
     "fair": 0.052632,
     "simulationOnly": [
      "pMutualPair",
      "pSingleLoop",
      "meanLoops"
     ]
    }
   ]
  },
  "loop": {
   "n": 10,
   "shuffles": 1000000,
   "pSingleLoop": 1,
   "pSomeoneSelf": 0,
   "pMutualPair": 0,
   "pFirstGivesToLast": 0.111608,
   "receiverOfFirst": [
    0.111338,
    0.111319,
    0.111078,
    0.110713,
    0.110938,
    0.110973,
    0.110574,
    0.111459,
    0.111608
   ],
   "couples": {
    "k": 3,
    "pNoCoupleAdjacent": 0.483925
   },
   "lastYear": {
    "pValid": 0.33016
   },
   "uniformity": {
    "people": 5,
    "shuffles": 240000,
    "loops": 24,
    "chiSquare": 17.7,
    "df": 23
   },
   "note": "Simulated presses of Shuffle; each name gives to the next, the last to the first. receiverOfFirst[j] = share in which person 1 gave to person j+2."
  },
  "helper": {
   "n": 10,
   "couples": 3,
   "sessions": 100000,
   "meanShuffles": 5.38952,
   "pLastGivesToFirst": 0.11595,
   "pairTypes": [
    {
     "type": "single to single",
     "giver": 10,
     "receiver": 7,
     "p": 0.10082
    },
    {
     "type": "single to someone with a partner",
     "giver": 10,
     "receiver": 1,
     "p": 0.11595
    },
    {
     "type": "someone with a partner to a single",
     "giver": 1,
     "receiver": 10,
     "p": 0.11499
    },
    {
     "type": "someone with a partner to a member of another couple",
     "giver": 1,
     "receiver": 3,
     "p": 0.13427
    }
   ],
   "pMutualPair": 0.41769,
   "uniformity": {
    "people": 6,
    "couples": 2,
    "sessions": 200000,
    "validDraws": 116,
    "chiSquare": 143.5,
    "df": 115
   },
   "note": "The widget's method: CueRandom.shuffle gives person i the i-th shuffled name; shuffle again until nobody has themselves or their partner. Couples are persons 1&2, 3&4, 5&6; persons 7-10 are single. pLastGivesToFirst = person 10 (single) gives to person 1 (has a partner). pairTypes = one example pair of each type; every value here is checked against theory.helper."
  }
 },
 "theory": {
  "hat": {
   "formulas": {
    "pWorks": "D(n)/n!, D(n) = derangements of n = n! × sum_{k=0}^{n} (-1)^k/k!; tends to 1/e = 0.367879...",
    "expectedTries": "n!/D(n) = average number of full draws of the hat INCLUDING the first one (geometric), tends to e = 2.718281...",
    "expectedRedraws": "expectedTries - 1 = average number of redraws AFTER the first draw, tends to e - 1 = 1.718281...",
    "pDoneWithin": "1 - (1 - D(n)/n!)^t",
    "expectedSelf": "exactly 1 for every n >= 1 (each of n people draws their own name with chance 1/n)",
    "pExactlyOneSelf": "n × D(n-1)/n!"
   },
   "oneOverE": 0.367879,
   "e": 2.718282,
   "note": "Exact values with BigInt. The hat on the wheel and a paper hat give the same chances: every way of handing out the names is equally likely (the wheel is only a model here: drawn in front of the group it is not secret). expectedTries counts full draws including the first (about 2.72); expectedRedraws counts the redraws after the first (about 1.72). gapTo1OverE = pWorks - 1/e (tiny from about 7 people on).",
   "rows": [
    {
     "people": 4,
     "derangements": "9",
     "permutations": "24",
     "pWorks": 0.375,
     "displayWorks": "37.5%",
     "fraction": "3/8",
     "pFails": 0.625,
     "displayFails": "62.5%",
     "expectedTries": 2.666667,
     "expectedTriesDisplay": "2.67",
     "expectedRedraws": 1.666667,
     "expectedRedrawsDisplay": "1.67",
     "pDoneWithin": {
      "1": 0.375,
      "2": 0.609375,
      "3": 0.755859,
      "5": 0.904633
     },
     "pMoreThan3Tries": 0.244141,
     "displayMoreThan3Tries": "24.4%",
     "pExactlyOneSelf": 0.333333,
     "displayExactlyOneSelf": "33.3%",
     "pTwoOrMoreSelf": 0.291667,
     "displayTwoOrMoreSelf": "29.2%",
     "expectedSelf": 1,
     "gapTo1OverE": 0.007121
    },
    {
     "people": 5,
     "derangements": "44",
     "permutations": "120",
     "pWorks": 0.366667,
     "displayWorks": "36.7%",
     "fraction": "11/30",
     "pFails": 0.633333,
     "displayFails": "63.3%",
     "expectedTries": 2.727273,
     "expectedTriesDisplay": "2.73",
     "expectedRedraws": 1.727273,
     "expectedRedrawsDisplay": "1.73",
     "pDoneWithin": {
      "1": 0.366667,
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     "displayFails": "63.2%",
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     "displayFails": "63.2%",
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    {
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   "widgetTable": [
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    {
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    {
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    {
     "people": 29,
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    {
     "people": 30,
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    {
     "people": 31,
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    {
     "people": 32,
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    {
     "people": 33,
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    {
     "people": 34,
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    {
     "people": 35,
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    {
     "people": 36,
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    {
     "people": 37,
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    {
     "people": 38,
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    {
     "people": 39,
     "pWorks": 0.367879441171,
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    {
     "people": 40,
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   ]
  },
  "uniformValid": {
   "formulas": {
    "pMutualPair": "1 - (permutations with every cycle of length >= 3)/D(n); tends to 1 - e^(-1/2) = 0.393469",
    "expectedMutualPairs": "C(n,2) D(n-2)/D(n), tends to 1/2",
    "pSpecificPairSwap": "D(n-2)/D(n): two named people draw each other",
    "pSingleLoop": "(n-1)!/D(n), about e/n",
    "expectedLoops": "sum over loop lengths k >= 2 of C(n,k)(k-1)! D(n-k)/D(n)",
    "pGiverToReceiver": "1/(n-1) exactly, by symmetry"
   },
   "limits": {
    "mutualPair": 0.393469,
    "expectedMutualPairs": 0.5
   },
   "note": "A uniformly random valid draw: every way of handing out the names in which nobody has themselves is equally likely. The redraw rule (a well-mixed hat redrawn in full after any failure) and the organizer's helper with no pairs to keep apart give this. With couples or last year's matches to avoid, the helper is uniform over the draws that respect those rules, which is a different distribution: there the giver-receiver chances are no longer all 1/(n-1) (see theory.helper). The loop method is different: it always gives one single loop and never a mutual pair, while every giver is still equally likely to get each other person (1/(n-1)).",
   "rows": [
    {
     "people": 4,
     "pMutualPair": 0.333333,
     "displayMutualPair": "33.3%",
     "expectedMutualPairs": 0.666667,
     "pSpecificPairSwap": 0.111111,
     "displaySpecificPairSwap": "11.1%",
     "pSingleLoop": 0.666667,
     "displaySingleLoop": "66.7%",
     "expectedLoops": 1.333333,
     "expectedLoopsDisplay": "1.33",
     "pGiverToReceiver": 0.333333,
     "fractionGiverToReceiver": "1/3",
     "displayGiverToReceiver": "33.3%"
    },
    {
     "people": 5,
     "pMutualPair": 0.454545,
     "displayMutualPair": "45.5%",
     "expectedMutualPairs": 0.454545,
     "pSpecificPairSwap": 0.045455,
     "displaySpecificPairSwap": "4.55%",
     "pSingleLoop": 0.545455,
     "displaySingleLoop": "54.5%",
     "expectedLoops": 1.454545,
     "expectedLoopsDisplay": "1.45",
     "pGiverToReceiver": 0.25,
     "fractionGiverToReceiver": "1/4",
     "displayGiverToReceiver": "25.0%"
    },
    {
     "people": 6,
     "pMutualPair": 0.396226,
     "displayMutualPair": "39.6%",
     "expectedMutualPairs": 0.509434,
     "pSpecificPairSwap": 0.033962,
     "displaySpecificPairSwap": "3.40%",
     "pSingleLoop": 0.45283,
     "displaySingleLoop": "45.3%",
     "expectedLoops": 1.603774,
     "expectedLoopsDisplay": "1.60",
     "pGiverToReceiver": 0.2,
     "fractionGiverToReceiver": "1/5",
     "displayGiverToReceiver": "20.0%"
    },
    {
     "people": 7,
     "pMutualPair": 0.385113,
     "displayMutualPair": "38.5%",
     "expectedMutualPairs": 0.498382,
     "pSpecificPairSwap": 0.023732,
     "displaySpecificPairSwap": "2.37%",
     "pSingleLoop": 0.38835,
     "displaySingleLoop": "38.8%",
     "expectedLoops": 1.724919,
     "expectedLoopsDisplay": "1.72",
     "pGiverToReceiver": 0.166667,
     "fractionGiverToReceiver": "1/6",
     "displayGiverToReceiver": "16.7%"
    },
    {
     "people": 8,
     "pMutualPair": 0.394054,
     "displayMutualPair": "39.4%",
     "expectedMutualPairs": 0.500236,
     "pSpecificPairSwap": 0.017866,
     "displaySpecificPairSwap": "1.79%",
     "pSingleLoop": 0.339783,
     "displaySingleLoop": "34.0%",
     "expectedLoops": 1.834828,
     "expectedLoopsDisplay": "1.83",
     "pGiverToReceiver": 0.142857,
     "fractionGiverToReceiver": "1/7",
     "displayGiverToReceiver": "14.3%"
    },
    {
     "people": 9,
     "pMutualPair": 0.394259,
     "displayMutualPair": "39.4%",
     "expectedMutualPairs": 0.49997,
     "pSpecificPairSwap": 0.013888,
     "displaySpecificPairSwap": "1.39%",
     "pSingleLoop": 0.302032,
     "displaySingleLoop": "30.2%",
     "expectedLoops": 1.933721,
     "expectedLoopsDisplay": "1.93",
     "pGiverToReceiver": 0.125,
     "fractionGiverToReceiver": "1/8",
     "displayGiverToReceiver": "12.5%"
    },
    {
     "people": 10,
     "pMutualPair": 0.393349,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.500003,
     "pSpecificPairSwap": 0.011111,
     "displaySpecificPairSwap": "1.11%",
     "pSingleLoop": 0.271828,
     "displaySingleLoop": "27.2%",
     "expectedLoops": 2.023832,
     "expectedLoopsDisplay": "2.02",
     "pGiverToReceiver": 0.111111,
     "fractionGiverToReceiver": "1/9",
     "displayGiverToReceiver": "11.1%"
    },
    {
     "people": 11,
     "pMutualPair": 0.393412,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.5,
     "pSpecificPairSwap": 0.009091,
     "displaySpecificPairSwap": "0.91%",
     "pSingleLoop": 0.247117,
     "displaySingleLoop": "24.7%",
     "expectedLoops": 2.106549,
     "expectedLoopsDisplay": "2.11",
     "pGiverToReceiver": 0.1,
     "fractionGiverToReceiver": "1/10",
     "displayGiverToReceiver": "10.0%"
    },
    {
     "people": 12,
     "pMutualPair": 0.393483,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.5,
     "pSpecificPairSwap": 0.007576,
     "displaySpecificPairSwap": "0.76%",
     "pSingleLoop": 0.226523,
     "displaySingleLoop": "22.7%",
     "expectedLoops": 2.18299,
     "expectedLoopsDisplay": "2.18",
     "pGiverToReceiver": 0.090909,
     "fractionGiverToReceiver": "1/11",
     "displayGiverToReceiver": "9.09%"
    },
    {
     "people": 13,
     "pMutualPair": 0.393473,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.5,
     "pSpecificPairSwap": 0.00641,
     "displaySpecificPairSwap": "0.64%",
     "pSingleLoop": 0.209099,
     "displaySingleLoop": "20.9%",
     "expectedLoops": 2.254033,
     "expectedLoopsDisplay": "2.25",
     "pGiverToReceiver": 0.083333,
     "fractionGiverToReceiver": "1/12",
     "displayGiverToReceiver": "8.33%"
    },
    {
     "people": 14,
     "pMutualPair": 0.393468,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.5,
     "pSpecificPairSwap": 0.005495,
     "displaySpecificPairSwap": "0.55%",
     "pSingleLoop": 0.194163,
     "displaySingleLoop": "19.4%",
     "expectedLoops": 2.320387,
     "expectedLoopsDisplay": "2.32",
     "pGiverToReceiver": 0.076923,
     "fractionGiverToReceiver": "1/13",
     "displayGiverToReceiver": "7.69%"
    },
    {
     "people": 15,
     "pMutualPair": 0.393469,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.5,
     "pSpecificPairSwap": 0.004762,
     "displaySpecificPairSwap": "0.48%",
     "pSingleLoop": 0.181219,
     "displaySingleLoop": "18.1%",
     "expectedLoops": 2.38263,
     "expectedLoopsDisplay": "2.38",
     "pGiverToReceiver": 0.071429,
     "fractionGiverToReceiver": "1/14",
     "displayGiverToReceiver": "7.14%"
    },
    {
     "people": 16,
     "pMutualPair": 0.393469,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.5,
     "pSpecificPairSwap": 0.004167,
     "displaySpecificPairSwap": "0.42%",
     "pSingleLoop": 0.169893,
     "displaySingleLoop": "17.0%",
     "expectedLoops": 2.441239,
     "expectedLoopsDisplay": "2.44",
     "pGiverToReceiver": 0.066667,
     "fractionGiverToReceiver": "1/15",
     "displayGiverToReceiver": "6.67%"
    },
    {
     "people": 17,
     "pMutualPair": 0.393469,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.5,
     "pSpecificPairSwap": 0.003676,
     "displaySpecificPairSwap": "0.37%",
     "pSingleLoop": 0.159899,
     "displaySingleLoop": "16.0%",
     "expectedLoops": 2.496615,
     "expectedLoopsDisplay": "2.50",
     "pGiverToReceiver": 0.0625,
     "fractionGiverToReceiver": "1/16",
     "displayGiverToReceiver": "6.25%"
    },
    {
     "people": 18,
     "pMutualPair": 0.393469,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.5,
     "pSpecificPairSwap": 0.003268,
     "displaySpecificPairSwap": "0.33%",
     "pSingleLoop": 0.151016,
     "displaySingleLoop": "15.1%",
     "expectedLoops": 2.549095,
     "expectedLoopsDisplay": "2.55",
     "pGiverToReceiver": 0.058824,
     "fractionGiverToReceiver": "1/17",
     "displayGiverToReceiver": "5.88%"
    },
    {
     "people": 19,
     "pMutualPair": 0.393469,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.5,
     "pSpecificPairSwap": 0.002924,
     "displaySpecificPairSwap": "0.29%",
     "pSingleLoop": 0.143067,
     "displaySingleLoop": "14.3%",
     "expectedLoops": 2.598964,
     "expectedLoopsDisplay": "2.60",
     "pGiverToReceiver": 0.055556,
     "fractionGiverToReceiver": "1/18",
     "displayGiverToReceiver": "5.56%"
    },
    {
     "people": 20,
     "pMutualPair": 0.393469,
     "displayMutualPair": "39.3%",
     "expectedMutualPairs": 0.5,
     "pSpecificPairSwap": 0.002632,
     "displaySpecificPairSwap": "0.26%",
     "pSingleLoop": 0.135914,
     "displaySingleLoop": "13.6%",
     "expectedLoops": 2.646471,
     "expectedLoopsDisplay": "2.65",
     "pGiverToReceiver": 0.052632,
     "fractionGiverToReceiver": "1/19",
     "displayGiverToReceiver": "5.26%"
    }
   ],
   "loopDistribution": {
    "6": [
     {
      "loops": 1,
      "p": 0.45283,
      "display": "45.3%"
     },
     {
      "loops": 2,
      "p": 0.490566,
      "display": "49.1%"
     },
     {
      "loops": 3,
      "p": 0.056604,
      "display": "5.66%"
     }
    ],
    "10": [
     {
      "loops": 1,
      "p": 0.271828,
      "display": "27.2%"
     },
     {
      "loops": 2,
      "p": 0.466962,
      "display": "46.7%"
     },
     {
      "loops": 3,
      "p": 0.227467,
      "display": "22.7%"
     },
     {
      "loops": 4,
      "p": 0.033035,
      "display": "3.30%"
     },
     {
      "loops": 5,
      "p": 0.000708,
      "display": "0.07%"
     }
    ],
    "20": [
     {
      "loops": 1,
      "p": 0.135914,
      "display": "13.6%"
     },
     {
      "loops": 2,
      "p": 0.33912,
      "display": "33.9%"
     },
     {
      "loops": 3,
      "p": 0.322509,
      "display": "32.3%"
     },
     {
      "loops": 4,
      "p": 0.154708,
      "display": "15.5%"
     },
     {
      "loops": 5,
      "p": 0.041075,
      "display": "4.11%"
     },
     {
      "loops": 6,
      "p": 0.006151,
      "display": "0.62%"
     },
     {
      "loops": 7,
      "p": 0.000501,
      "display": "0.05%"
     },
     {
      "loops": 8,
      "p": 0.00002,
      "display": "<0.01%"
     }
    ]
   },
   "uniformityPeople": 5,
   "uniformityValidDraws": 44
  },
  "putBack": {
   "rule": "a person who draws their own name puts it back and draws again (Spin again); if the last person is left with only their own name the draw dead-ends and starts over from the top",
   "formulas": {
    "pDeadEnd": "exact dynamic program over f = names still in the hat whose owners have not drawn yet (by symmetry every such set of a given size is equally likely); P(dead end) = P(the last person's own name is the last one left)",
    "expectedPutBacksPerAttempt": "sum over draws with m >= 2 names left of P(own name still in the hat) × 1/(m-1)",
    "bias": "exhaustive enumeration of every complete draw (4-10 people); the chance of one complete draw is the product of 1/(names the person could take) over the people, and draws that dead-end are started again, so the result is that product divided by P(no dead end)",
    "chain": "exact Markov chain over the set of names still in the hat (4-20 people), integer weights over the common denominator prod_{m=2..n} m(m-1); it reproduces the DP's dead-end chance at every size and the enumeration's pairing chances for 4-10 people"
   },
   "note": "Exact fractions (BigInt). People are numbered in the order they draw (1 draws first). 'fair' = 1/(n-1), what every giver-receiver pair gets in a uniformly random valid draw and in the loop. ratioMatrix[a][b] = P(person a+1 gives to person b+1)/fair, 0 on the diagonal. lastTwoSwap = P(the last two people to draw get each other); lastTwoSwapUniform is the same in a uniformly random valid draw. mostLikelyPair/leastLikelyPair give one pair; tiedPairs = how many giver-receiver pairs share that exact chance (at 4 people two pairs tie for least likely, so do not name a unique one). pMutualPair, pSingleLoop and expectedLoops barely change under the put-back rule but are not equal to the fair values (the *Diff fields are exact differences, put-back minus fair). wholeDrawMostOverLeast = the most likely complete draw's chance divided by the least likely one's: exactly n-1 for 4-10 people (checked only in that enumerated range, no general proof); wholeDraw*OverFair compare with 1/D(n). totalVariation = the share of probability that would have to move to make every valid draw equally likely. dpRows covers 4-20 people; biasRows 4-10 (enumeration); chainRows 4-20 (exact chain: the last person's row, person 1 to person n, the last two swapping).",
   "dpRows": [
    {
     "people": 4,
     "pDeadEnd": 0.138889,
     "displayDeadEnd": "13.9%",
     "deadEndOneIn": "1 in 7.2",
     "fraction": "5/36",
     "pDeadEndTimesN": 0.5556,
     "expectedPutBacksPerAttempt": 1.055556,
     "expectedAttempts": 1.16129,
     "pDeadEndVsHatFails": 0.2222
    },
    {
     "people": 5,
     "pDeadEnd": 0.131944,
     "displayDeadEnd": "13.2%",
     "deadEndOneIn": "1 in 7.6",
     "fraction": "19/144",
     "pDeadEndTimesN": 0.6597,
     "expectedPutBacksPerAttempt": 1.072917,
     "expectedAttempts": 1.152,
     "pDeadEndVsHatFails": 0.2083
    },
    {
     "people": 6,
     "pDeadEnd": 0.112778,
     "displayDeadEnd": "11.3%",
     "deadEndOneIn": "1 in 8.9",
     "fraction": "203/1800",
     "pDeadEndTimesN": 0.6767,
     "expectedPutBacksPerAttempt": 1.082917,
     "expectedAttempts": 1.127113,
     "pDeadEndVsHatFails": 0.1785
    },
    {
     "people": 7,
     "pDeadEnd": 0.100532,
     "displayDeadEnd": "10.1%",
     "deadEndOneIn": "1 in 9.9",
     "fraction": "4343/43200",
     "pDeadEndTimesN": 0.7037,
     "expectedPutBacksPerAttempt": 1.087778,
     "expectedAttempts": 1.111769,
     "pDeadEndVsHatFails": 0.159
    },
    {
     "people": 8,
     "pDeadEnd": 0.090495,
     "displayDeadEnd": "9.05%",
     "deadEndOneIn": "1 in 11",
     "fraction": "63853/705600",
     "pDeadEndTimesN": 0.724,
     "expectedPutBacksPerAttempt": 1.089989,
     "expectedAttempts": 1.099499,
     "pDeadEndVsHatFails": 0.1432
    },
    {
     "people": 9,
     "pDeadEnd": 0.082382,
     "displayDeadEnd": "8.24%",
     "deadEndOneIn": "1 in 12",
     "fraction": "58129/705600",
     "pDeadEndTimesN": 0.7414,
     "expectedPutBacksPerAttempt": 1.090647,
     "expectedAttempts": 1.089779,
     "pDeadEndVsHatFails": 0.1303
    },
    {
     "people": 10,
     "pDeadEnd": 0.075646,
     "displayDeadEnd": "7.56%",
     "deadEndOneIn": "1 in 13",
     "fraction": "160127/2116800",
     "pDeadEndTimesN": 0.7565,
     "expectedPutBacksPerAttempt": 1.09038,
     "expectedAttempts": 1.081836,
     "pDeadEndVsHatFails": 0.1197
    },
    {
     "people": 11,
     "pDeadEnd": 0.06996,
     "displayDeadEnd": "7.00%",
     "deadEndOneIn": "1 in 14",
     "fraction": "8885501/127008000",
     "pDeadEndTimesN": 0.7696,
     "expectedPutBacksPerAttempt": 1.08955,
     "expectedAttempts": 1.075223,
     "pDeadEndVsHatFails": 0.1107
    },
    {
     "people": 12,
     "pDeadEnd": 0.065093,
     "displayDeadEnd": "6.51%",
     "deadEndOneIn": "1 in 15",
     "fraction": "1500518539/23051952000",
     "pDeadEndTimesN": 0.7811,
     "expectedPutBacksPerAttempt": 1.088377,
     "expectedAttempts": 1.069625,
     "pDeadEndVsHatFails": 0.103
    },
    {
     "people": 13,
     "pDeadEnd": 0.060876,
     "displayDeadEnd": "6.09%",
     "deadEndOneIn": "1 in 16",
     "fraction": "404156337271/6638962176000",
     "pDeadEndTimesN": 0.7914,
     "expectedPutBacksPerAttempt": 1.087,
     "expectedAttempts": 1.064823,
     "pDeadEndVsHatFails": 0.0963
    },
    {
     "people": 14,
     "pDeadEnd": 0.057186,
     "displayDeadEnd": "5.72%",
     "deadEndOneIn": "1 in 17",
     "fraction": "16040576541971/280496151936000",
     "pDeadEndTimesN": 0.8006,
     "expectedPutBacksPerAttempt": 1.085507,
     "expectedAttempts": 1.060655,
     "pDeadEndVsHatFails": 0.0905
    },
    {
     "people": 15,
     "pDeadEnd": 0.053929,
     "displayDeadEnd": "5.39%",
     "deadEndOneIn": "1 in 19",
     "fraction": null,
     "pDeadEndTimesN": 0.8089,
     "expectedPutBacksPerAttempt": 1.083955,
     "expectedAttempts": 1.057003,
     "pDeadEndVsHatFails": 0.0853
    },
    {
     "people": 16,
     "pDeadEnd": 0.05103,
     "displayDeadEnd": "5.10%",
     "deadEndOneIn": "1 in 20",
     "fraction": null,
     "pDeadEndTimesN": 0.8165,
     "expectedPutBacksPerAttempt": 1.082381,
     "expectedAttempts": 1.053775,
     "pDeadEndVsHatFails": 0.0807
    },
    {
     "people": 17,
     "pDeadEnd": 0.048434,
     "displayDeadEnd": "4.84%",
     "deadEndOneIn": "1 in 21",
     "fraction": null,
     "pDeadEndTimesN": 0.8234,
     "expectedPutBacksPerAttempt": 1.080812,
     "expectedAttempts": 1.0509,
     "pDeadEndVsHatFails": 0.0766
    },
    {
     "people": 18,
     "pDeadEnd": 0.046095,
     "displayDeadEnd": "4.61%",
     "deadEndOneIn": "1 in 22",
     "fraction": null,
     "pDeadEndTimesN": 0.8297,
     "expectedPutBacksPerAttempt": 1.079262,
     "expectedAttempts": 1.048323,
     "pDeadEndVsHatFails": 0.0729
    },
    {
     "people": 19,
     "pDeadEnd": 0.043976,
     "displayDeadEnd": "4.40%",
     "deadEndOneIn": "1 in 23",
     "fraction": null,
     "pDeadEndTimesN": 0.8355,
     "expectedPutBacksPerAttempt": 1.077743,
     "expectedAttempts": 1.045999,
     "pDeadEndVsHatFails": 0.0696
    },
    {
     "people": 20,
     "pDeadEnd": 0.042047,
     "displayDeadEnd": "4.20%",
     "deadEndOneIn": "1 in 24",
     "fraction": null,
     "pDeadEndTimesN": 0.8409,
     "expectedPutBacksPerAttempt": 1.076262,
     "expectedAttempts": 1.043892,
     "pDeadEndVsHatFails": 0.0665
    }
   ],
   "biasRows": [
    {
     "people": 4,
     "validDraws": "9",
     "fair": 0.333333,
     "fairFraction": "1/3",
     "fairDisplay": "33.3%",
     "lastGivesToPrevious": 0.451613,
     "lastGivesToPreviousDisplay": "45.2%",
     "lastGivesToPreviousRatio": 1.3548,
     "lastGivesToFirst": 0.258065,
     "lastGivesToFirstDisplay": "25.8%",
     "lastGivesToFirstRatio": 0.7742,
     "firstGivesToLast": 0.387097,
     "firstGivesToLastDisplay": "38.7%",
     "firstGivesToLastRatio": 1.1613,
     "mostLikelyPair": {
      "giver": 4,
      "receiver": 3,
      "p": 0.451613,
      "display": "45.2%",
      "ratio": 1.3548,
      "tiedPairs": 1
     },
     "leastLikelyPair": {
      "giver": 2,
      "receiver": 3,
      "p": 0.258065,
      "display": "25.8%",
      "ratio": 0.7742,
      "tiedPairs": 2
     },
     "lastTwoSwap": 0.129032,
     "lastTwoSwapDisplay": "12.9%",
     "lastTwoSwapUniform": 0.111111,
     "lastTwoSwapUniformDisplay": "11.1%",
     "lastTwoSwapRatio": 1.1613,
     "pMutualPair": 0.322581,
     "pMutualPairUniform": 0.333333,
     "pMutualPairDiff": -0.010753,
     "pMutualPairDisplay": "32.3%",
     "pMutualPairUniformDisplay": "33.3%",
     "pSingleLoop": 0.677419,
     "pSingleLoopUniform": 0.666667,
     "pSingleLoopDiff": 0.010753,
     "pSingleLoopDisplay": "67.7%",
     "pSingleLoopUniformDisplay": "66.7%",
     "expectedLoops": 1.322581,
     "expectedLoopsUniform": 1.333333,
     "expectedLoopsDiff": -0.010753,
     "wholeDrawMostOverLeast": 3,
     "wholeDrawMostOverFair": 1.7419,
     "wholeDrawLeastOverFair": 0.5806,
     "shareOfDrawsMoreLikelyThanFair": 0.3333,
     "totalVariation": 0.11828,
     "totalVariationDisplay": "11.8%",
     "lastRow": [
      {
       "receiver": 1,
       "p": 0.258065,
       "display": "25.8%",
       "ratio": 0.7742
      },
      {
       "receiver": 2,
       "p": 0.290323,
       "display": "29.0%",
       "ratio": 0.871
      },
      {
       "receiver": 3,
       "p": 0.451613,
       "display": "45.2%",
       "ratio": 1.3548
      }
     ],
     "ratioMatrix": [
      [
       0,
       0.9677,
       0.871,
       1.1613
      ],
      [
       1.2581,
       0,
       0.7742,
       0.9677
      ],
      [
       0.9677,
       1.1613,
       0,
       0.871
      ],
      [
       0.7742,
       0.871,
       1.3548,
       0
      ]
     ]
    },
    {
     "people": 5,
     "validDraws": "44",
     "fair": 0.25,
     "fairFraction": "1/4",
     "fairDisplay": "25.0%",
     "lastGivesToPrevious": 0.36,
     "lastGivesToPreviousDisplay": "36.0%",
     "lastGivesToPreviousRatio": 1.44,
     "lastGivesToFirst": 0.192,
     "lastGivesToFirstDisplay": "19.2%",
     "lastGivesToFirstRatio": 0.768,
     "firstGivesToLast": 0.288,
     "firstGivesToLastDisplay": "28.8%",
     "firstGivesToLastRatio": 1.152,
     "mostLikelyPair": {
      "giver": 5,
      "receiver": 4,
      "p": 0.36,
      "display": "36.0%",
      "ratio": 1.44,
      "tiedPairs": 1
     },
     "leastLikelyPair": {
      "giver": 5,
      "receiver": 1,
      "p": 0.192,
      "display": "19.2%",
      "ratio": 0.768,
      "tiedPairs": 1
     },
     "lastTwoSwap": 0.056,
     "lastTwoSwapDisplay": "5.60%",
     "lastTwoSwapUniform": 0.045455,
     "lastTwoSwapUniformDisplay": "4.55%",
     "lastTwoSwapRatio": 1.232,
     "pMutualPair": 0.45,
     "pMutualPairUniform": 0.454545,
     "pMutualPairDiff": -0.004545,
     "pMutualPairDisplay": "45.0%",
     "pMutualPairUniformDisplay": "45.5%",
     "pSingleLoop": 0.55,
     "pSingleLoopUniform": 0.545455,
     "pSingleLoopDiff": 0.004545,
     "pSingleLoopDisplay": "55.0%",
     "pSingleLoopUniformDisplay": "54.5%",
     "expectedLoops": 1.45,
     "expectedLoopsUniform": 1.454545,
     "expectedLoopsDiff": -0.004545,
     "wholeDrawMostOverLeast": 4,
     "wholeDrawMostOverFair": 2.112,
     "wholeDrawLeastOverFair": 0.528,
     "shareOfDrawsMoreLikelyThanFair": 0.5455,
     "totalVariation": 0.126545,
     "totalVariationDisplay": "12.7%",
     "lastRow": [
      {
       "receiver": 1,
       "p": 0.192,
       "display": "19.2%",
       "ratio": 0.768
      },
      {
       "receiver": 2,
       "p": 0.208,
       "display": "20.8%",
       "ratio": 0.832
      },
      {
       "receiver": 3,
       "p": 0.24,
       "display": "24.0%",
       "ratio": 0.96
      },
      {
       "receiver": 4,
       "p": 0.36,
       "display": "36.0%",
       "ratio": 1.44
      }
     ],
     "ratioMatrix": [
      [
       0,
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  "loop": {
   "formulas": {
    "loops": "(n-1)! possible loops, each produced by n of the n! shuffles (one per starting point), so each loop has chance 1/(n-1)!",
    "giverToReceiver": "1/(n-1) for every giver and receiver",
    "pSingleLoop": "1 (always)",
    "pMutualPair": "0 for n >= 3",
    "pSelf": "0"
   },
   "note": "Shuffle in app.js calls CueRandom.shuffle (Fisher–Yates) on the non-empty lines of the Names box, so every order is equally likely. Reading the list as a loop gives every person a giver and a receiver, never themselves, never a mutual pair. Whoever presses Shuffle sees the whole list, and so every pairing.",
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     "possibleValidDraws": "9",
     "shareOfValidDrawsThatAreLoops": 0.666667
    },
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     "shareOfValidDrawsThatAreLoops": 0.545455
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     "shareOfValidDrawsThatAreLoops": 0.38835
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     "people": 8,
     "possibleLoops": "5040",
     "possibleValidDraws": "14833",
     "shareOfValidDrawsThatAreLoops": 0.339783
    },
    {
     "people": 9,
     "possibleLoops": "40320",
     "possibleValidDraws": "133496",
     "shareOfValidDrawsThatAreLoops": 0.302032
    },
    {
     "people": 10,
     "possibleLoops": "362880",
     "possibleValidDraws": "1334961",
     "shareOfValidDrawsThatAreLoops": 0.271828
    },
    {
     "people": 11,
     "possibleLoops": "3628800",
     "possibleValidDraws": "14684570",
     "shareOfValidDrawsThatAreLoops": 0.247117
    },
    {
     "people": 12,
     "possibleLoops": "39916800",
     "possibleValidDraws": "176214841",
     "shareOfValidDrawsThatAreLoops": 0.226523
    },
    {
     "people": 13,
     "possibleLoops": "479001600",
     "possibleValidDraws": "2290792932",
     "shareOfValidDrawsThatAreLoops": 0.209099
    },
    {
     "people": 14,
     "possibleLoops": "6227020800",
     "possibleValidDraws": "32071101049",
     "shareOfValidDrawsThatAreLoops": 0.194163
    },
    {
     "people": 15,
     "possibleLoops": "87178291200",
     "possibleValidDraws": "481066515734",
     "shareOfValidDrawsThatAreLoops": 0.181219
    },
    {
     "people": 16,
     "possibleLoops": "1307674368000",
     "possibleValidDraws": "7697064251745",
     "shareOfValidDrawsThatAreLoops": 0.169893
    },
    {
     "people": 17,
     "possibleLoops": "20922789888000",
     "possibleValidDraws": "130850092279664",
     "shareOfValidDrawsThatAreLoops": 0.159899
    },
    {
     "people": 18,
     "possibleLoops": "355687428096000",
     "possibleValidDraws": "2355301661033953",
     "shareOfValidDrawsThatAreLoops": 0.151016
    },
    {
     "people": 19,
     "possibleLoops": "6402373705728000",
     "possibleValidDraws": "44750731559645106",
     "shareOfValidDrawsThatAreLoops": 0.143067
    },
    {
     "people": 20,
     "possibleLoops": "121645100408832000",
     "possibleValidDraws": "895014631192902121",
     "shareOfValidDrawsThatAreLoops": 0.135914
    }
   ]
  },
  "couples": {
   "rule": "k couples (persons 1&2, 3&4, ...) who must not draw each other, on top of nobody drawing themselves",
   "formulas": {
    "validDraws": "inclusion–exclusion with the rook polynomial (1 + 4x + 2x^2)^k (1 + x)^(n-2k): sum_j (-1)^j r_j (n-j)!",
    "validLoops": "sum_j (-1)^j C(k,j) 2^j (n-j-1)!: loops with no couple next to each other (glue j couples, either order)",
    "expectedHatTries": "n!/validDraws full draws of the hat (or shuffles in the helper) until nobody has themselves or their partner",
    "expectedShuffles": "(n-1)!/validLoops presses of Shuffle until no couple sits next to each other in the loop"
   },
   "note": "Exact BigInt counts, cross-checked by brute force for n <= 8. pNoSelfDrawBreaksCouples = share of draws with nobody drawing themselves in which some couple draws each other. In the loop, a couple breaks the rule when the two names are next to each other in the shuffled list (first and last count as next to each other).",
   "rows": [
    {
     "people": 4,
     "couples": 0,
     "validDraws": "9",
     "pHatValid": 0.375,
     "displayHatValid": "37.5%",
     "expectedHatTries": 2.666667,
     "expectedHatTriesDisplay": "2.67",
     "pNoSelfDrawBreaksCouples": 0,
     "displayNoSelfDrawBreaksCouples": "0%",
     "validLoops": "6",
     "pLoopValid": 1,
     "displayLoopValid": "100%",
     "expectedShuffles": 1,
     "expectedShufflesDisplay": "1.00"
    },
    {
     "people": 4,
     "couples": 1,
     "validDraws": "4",
     "pHatValid": 0.166667,
     "displayHatValid": "16.7%",
     "expectedHatTries": 6,
     "expectedHatTriesDisplay": "6.00",
     "pNoSelfDrawBreaksCouples": 0.555556,
     "displayNoSelfDrawBreaksCouples": "55.6%",
     "validLoops": "2",
     "pLoopValid": 0.333333,
     "displayLoopValid": "33.3%",
     "expectedShuffles": 3,
     "expectedShufflesDisplay": "3.00"
    },
    {
     "people": 4,
     "couples": 2,
     "validDraws": "4",
     "pHatValid": 0.166667,
     "displayHatValid": "16.7%",
     "expectedHatTries": 6,
     "expectedHatTriesDisplay": "6.00",
     "pNoSelfDrawBreaksCouples": 0.555556,
     "displayNoSelfDrawBreaksCouples": "55.6%",
     "validLoops": "2",
     "pLoopValid": 0.333333,
     "displayLoopValid": "33.3%",
     "expectedShuffles": 3,
     "expectedShufflesDisplay": "3.00"
    },
    {
     "people": 5,
     "couples": 0,
     "validDraws": "44",
     "pHatValid": 0.366667,
     "displayHatValid": "36.7%",
     "expectedHatTries": 2.727273,
     "expectedHatTriesDisplay": "2.73",
     "pNoSelfDrawBreaksCouples": 0,
     "displayNoSelfDrawBreaksCouples": "0%",
     "validLoops": "24",
     "pLoopValid": 1,
     "displayLoopValid": "100%",
     "expectedShuffles": 1,
     "expectedShufflesDisplay": "1.00"
    },
    {
     "people": 5,
     "couples": 1,
     "validDraws": "24",
     "pHatValid": 0.2,
     "displayHatValid": "20.0%",
     "expectedHatTries": 5,
     "expectedHatTriesDisplay": "5.00",
     "pNoSelfDrawBreaksCouples": 0.454545,
     "displayNoSelfDrawBreaksCouples": "45.5%",
     "validLoops": "12",
     "pLoopValid": 0.5,
     "displayLoopValid": "50.0%",
     "expectedShuffles": 2,
     "expectedShufflesDisplay": "2.00"
    },
    {
     "people": 5,
     "couples": 2,
     "validDraws": "16",
     "pHatValid": 0.133333,
     "displayHatValid": "13.3%",
     "expectedHatTries": 7.5,
     "expectedHatTriesDisplay": "7.50",
     "pNoSelfDrawBreaksCouples": 0.636364,
     "displayNoSelfDrawBreaksCouples": "63.6%",
     "validLoops": "8",
     "pLoopValid": 0.333333,
     "displayLoopValid": "33.3%",
     "expectedShuffles": 3,
     "expectedShufflesDisplay": "3.00"
    },
    {
     "people": 6,
     "couples": 0,
     "validDraws": "265",
     "pHatValid": 0.368056,
     "displayHatValid": "36.8%",
     "expectedHatTries": 2.716981,
     "expectedHatTriesDisplay": "2.72",
     "pNoSelfDrawBreaksCouples": 0,
     "displayNoSelfDrawBreaksCouples": "0%",
     "validLoops": "120",
     "pLoopValid": 1,
     "displayLoopValid": "100%",
     "expectedShuffles": 1,
     "expectedShufflesDisplay": "1.00"
    },
    {
     "people": 6,
     "couples": 1,
     "validDraws": "168",
     "pHatValid": 0.233333,
     "displayHatValid": "23.3%",
     "expectedHatTries": 4.285714,
     "expectedHatTriesDisplay": "4.29",
     "pNoSelfDrawBreaksCouples": 0.366038,
     "displayNoSelfDrawBreaksCouples": "36.6%",
     "validLoops": "72",
     "pLoopValid": 0.6,
     "displayLoopValid": "60.0%",
     "expectedShuffles": 1.666667,
     "expectedShufflesDisplay": "1.67"
    },
    {
     "people": 6,
     "couples": 2,
     "validDraws": "116",
     "pHatValid": 0.161111,
     "displayHatValid": "16.1%",
     "expectedHatTries": 6.206897,
     "expectedHatTriesDisplay": "6.21",
     "pNoSelfDrawBreaksCouples": 0.562264,
     "displayNoSelfDrawBreaksCouples": "56.2%",
     "validLoops": "48",
     "pLoopValid": 0.4,
     "displayLoopValid": "40.0%",
     "expectedShuffles": 2.5,
     "expectedShufflesDisplay": "2.50"
    },
    {
     "people": 6,
     "couples": 3,
     "validDraws": "80",
     "pHatValid": 0.111111,
     "displayHatValid": "11.1%",
     "expectedHatTries": 9,
     "expectedHatTriesDisplay": "9.00",
     "pNoSelfDrawBreaksCouples": 0.698113,
     "displayNoSelfDrawBreaksCouples": "69.8%",
     "validLoops": "32",
     "pLoopValid": 0.266667,
     "displayLoopValid": "26.7%",
     "expectedShuffles": 3.75,
     "expectedShufflesDisplay": "3.75"
    },
    {
     "people": 7,
     "couples": 0,
     "validDraws": "1854",
     "pHatValid": 0.367857,
     "displayHatValid": "36.8%",
     "expectedHatTries": 2.718447,
     "expectedHatTriesDisplay": "2.72",
     "pNoSelfDrawBreaksCouples": 0,
     "displayNoSelfDrawBreaksCouples": "0%",
     "validLoops": "720",
     "pLoopValid": 1,
     "displayLoopValid": "100%",
     "expectedShuffles": 1,
     "expectedShufflesDisplay": "1.00"
    },
    {
     "people": 7,
     "couples": 1,
     "validDraws": "1280",
     "pHatValid": 0.253968,
     "displayHatValid": "25.4%",
     "expectedHatTries": 3.9375,
     "expectedHatTriesDisplay": "3.94",
     "pNoSelfDrawBreaksCouples": 0.309601,
     "displayNoSelfDrawBreaksCouples": "31.0%",
     "validLoops": "480",
     "pLoopValid": 0.666667,
     "displayLoopValid": "66.7%",
     "expectedShuffles": 1.5,
     "expectedShufflesDisplay": "1.50"
    },
    {
     "people": 7,
     "couples": 2,
     "validDraws": "920",
     "pHatValid": 0.18254,
     "displayHatValid": "18.3%",
     "expectedHatTries": 5.478261,
     "expectedHatTriesDisplay": "5.48",
     "pNoSelfDrawBreaksCouples": 0.503776,
     "displayNoSelfDrawBreaksCouples": "50.4%",
     "validLoops": "336",
     "pLoopValid": 0.466667,
     "displayLoopValid": "46.7%",
     "expectedShuffles": 2.142857,
     "expectedShufflesDisplay": "2.14"
    },
    {
     "people": 7,
     "couples": 3,
     "validDraws": "672",
     "pHatValid": 0.133333,
     "displayHatValid": "13.3%",
     "expectedHatTries": 7.5,
     "expectedHatTriesDisplay": "7.50",
     "pNoSelfDrawBreaksCouples": 0.63754,
     "displayNoSelfDrawBreaksCouples": "63.8%",
     "validLoops": "240",
     "pLoopValid": 0.333333,
     "displayLoopValid": "33.3%",
     "expectedShuffles": 3,
     "expectedShufflesDisplay": "3.00"
    },
    {
     "people": 8,
     "couples": 0,
     "validDraws": "14833",
     "pHatValid": 0.367882,
     "displayHatValid": "36.8%",
     "expectedHatTries": 2.718263,
     "expectedHatTriesDisplay": "2.72",
     "pNoSelfDrawBreaksCouples": 0,
     "displayNoSelfDrawBreaksCouples": "0%",
     "validLoops": "5040",
     "pLoopValid": 1,
     "displayLoopValid": "100%",
     "expectedShuffles": 1,
     "expectedShufflesDisplay": "1.00"
    },
    {
     "people": 8,
     "couples": 1,
     "validDraws": "10860",
     "pHatValid": 0.269345,
     "displayHatValid": "26.9%",
     "expectedHatTries": 3.712707,
     "expectedHatTriesDisplay": "3.71",
     "pNoSelfDrawBreaksCouples": 0.267849,
     "displayNoSelfDrawBreaksCouples": "26.8%",
     "validLoops": "3600",
     "pLoopValid": 0.714286,
     "displayLoopValid": "71.4%",
     "expectedShuffles": 1.4,
     "expectedShufflesDisplay": "1.40"
    },
    {
     "people": 8,
     "couples": 2,
     "validDraws": "8132",
     "pHatValid": 0.201687,
     "displayHatValid": "20.2%",
     "expectedHatTries": 4.95819,
     "expectedHatTriesDisplay": "4.96",
     "pNoSelfDrawBreaksCouples": 0.451763,
     "displayNoSelfDrawBreaksCouples": "45.2%",
     "validLoops": "2640",
     "pLoopValid": 0.52381,
     "displayLoopValid": "52.4%",
     "expectedShuffles": 1.909091,
     "expectedShufflesDisplay": "1.91"
    },
    {
     "people": 8,
     "couples": 3,
     "validDraws": "6176",
     "pHatValid": 0.153175,
     "displayHatValid": "15.3%",
     "expectedHatTries": 6.528497,
     "expectedHatTriesDisplay": "6.53",
     "pNoSelfDrawBreaksCouples": 0.583631,
     "displayNoSelfDrawBreaksCouples": "58.4%",
     "validLoops": "1968",
     "pLoopValid": 0.390476,
     "displayLoopValid": "39.0%",
     "expectedShuffles": 2.560976,
     "expectedShufflesDisplay": "2.56"
    },
    {
     "people": 8,
     "couples": 4,
     "validDraws": "4752",
     "pHatValid": 0.117857,
     "displayHatValid": "11.8%",
     "expectedHatTries": 8.484848,
     "expectedHatTriesDisplay": "8.48",
     "pNoSelfDrawBreaksCouples": 0.679633,
     "displayNoSelfDrawBreaksCouples": "68.0%",
     "validLoops": "1488",
     "pLoopValid": 0.295238,
     "displayLoopValid": "29.5%",
     "expectedShuffles": 3.387097,
     "expectedShufflesDisplay": "3.39"
    },
    {
     "people": 9,
     "couples": 0,
     "validDraws": "133496",
     "pHatValid": 0.367879,
     "displayHatValid": "36.8%",
     "expectedHatTries": 2.718284,
     "expectedHatTriesDisplay": "2.72",
     "pNoSelfDrawBreaksCouples": 0,
     "displayNoSelfDrawBreaksCouples": "0%",
     "validLoops": "40320",
     "pLoopValid": 1,
     "displayLoopValid": "100%",
     "expectedShuffles": 1,
     "expectedShufflesDisplay": "1.00"
    },
    {
     "people": 9,
     "couples": 1,
     "validDraws": "101976",
     "pHatValid": 0.281019,
     "displayHatValid": "28.1%",
     "expectedHatTries": 3.558484,
     "expectedHatTriesDisplay": "3.56",
     "pNoSelfDrawBreaksCouples": 0.236112,
     "displayNoSelfDrawBreaksCouples": "23.6%",
     "validLoops": "30240",
     "pLoopValid": 0.75,
     "displayLoopValid": "75.0%",
     "expectedShuffles": 1.333333,
     "expectedShufflesDisplay": "1.33"
    },
    {
     "people": 9,
     "couples": 2,
     "validDraws": "78976",
     "pHatValid": 0.217637,
     "displayHatValid": "21.8%",
     "expectedHatTries": 4.594814,
     "expectedHatTriesDisplay": "4.59",
     "pNoSelfDrawBreaksCouples": 0.408402,
     "displayNoSelfDrawBreaksCouples": "40.8%",
     "validLoops": "23040",
     "pLoopValid": 0.571429,
     "displayLoopValid": "57.1%",
     "expectedShuffles": 1.75,
     "expectedShufflesDisplay": "1.75"
    },
    {
     "people": 9,
     "couples": 3,
     "validDraws": "61792",
     "pHatValid": 0.170282,
     "displayHatValid": "17.0%",
     "expectedHatTries": 5.872605,
     "expectedHatTriesDisplay": "5.87",
     "pNoSelfDrawBreaksCouples": 0.537125,
     "displayNoSelfDrawBreaksCouples": "53.7%",
     "validLoops": "17760",
     "pLoopValid": 0.440476,
     "displayLoopValid": "44.0%",
     "expectedShuffles": 2.27027,
     "expectedShufflesDisplay": "2.27"
    },
    {
     "people": 9,
     "couples": 4,
     "validDraws": "48768",
     "pHatValid": 0.134392,
     "displayHatValid": "13.4%",
     "expectedHatTries": 7.440945,
     "expectedHatTriesDisplay": "7.44",
     "pNoSelfDrawBreaksCouples": 0.634686,
     "displayNoSelfDrawBreaksCouples": "63.5%",
     "validLoops": "13824",
     "pLoopValid": 0.342857,
     "displayLoopValid": "34.3%",
     "expectedShuffles": 2.916667,
     "expectedShufflesDisplay": "2.92"
    },
    {
     "people": 10,
     "couples": 0,
     "validDraws": "1334961",
     "pHatValid": 0.367879,
     "displayHatValid": "36.8%",
     "expectedHatTries": 2.718282,
     "expectedHatTriesDisplay": "2.72",
     "pNoSelfDrawBreaksCouples": 0,
     "displayNoSelfDrawBreaksCouples": "0%",
     "validLoops": "362880",
     "pLoopValid": 1,
     "displayLoopValid": "100%",
     "expectedShuffles": 1,
     "expectedShufflesDisplay": "1.00"
    },
    {
     "people": 10,
     "couples": 1,
     "validDraws": "1053136",
     "pHatValid": 0.290216,
     "displayHatValid": "29.0%",
     "expectedHatTries": 3.445709,
     "expectedHatTriesDisplay": "3.45",
     "pNoSelfDrawBreaksCouples": 0.211111,
     "displayNoSelfDrawBreaksCouples": "21.1%",
     "validLoops": "282240",
     "pLoopValid": 0.777778,
     "displayLoopValid": "77.8%",
     "expectedShuffles": 1.285714,
     "expectedShufflesDisplay": "1.29"
    },
    {
     "people": 10,
     "couples": 2,
     "validDraws": "838324",
     "pHatValid": 0.23102,
     "displayHatValid": "23.1%",
     "expectedHatTries": 4.328637,
     "expectedHatTriesDisplay": "4.33",
     "pNoSelfDrawBreaksCouples": 0.372024,
     "displayNoSelfDrawBreaksCouples": "37.2%",
     "validLoops": "221760",
     "pLoopValid": 0.611111,
     "displayLoopValid": "61.1%",
     "expectedShuffles": 1.636364,
     "expectedShufflesDisplay": "1.64"
    },
    {
     "people": 10,
     "couples": 3,
     "validDraws": "672240",
     "pHatValid": 0.185251,
     "displayHatValid": "18.5%",
     "expectedHatTries": 5.398072,
     "expectedHatTriesDisplay": "5.40",
     "pNoSelfDrawBreaksCouples": 0.496435,
     "displayNoSelfDrawBreaksCouples": "49.6%",
     "validLoops": "175680",
     "pLoopValid": 0.484127,
     "displayLoopValid": "48.4%",
     "expectedShuffles": 2.065574,
     "expectedShufflesDisplay": "2.07"
    },
    {
     "people": 10,
     "couples": 4,
     "validDraws": "542480",
     "pHatValid": 0.149493,
     "displayHatValid": "14.9%",
     "expectedHatTries": 6.689279,
     "expectedHatTriesDisplay": "6.69",
     "pNoSelfDrawBreaksCouples": 0.593636,
     "displayNoSelfDrawBreaksCouples": "59.4%",
     "validLoops": "140160",
     "pLoopValid": 0.386243,
     "displayLoopValid": "38.6%",
     "expectedShuffles": 2.589041,
     "expectedShufflesDisplay": "2.59"
    },
    {
     "people": 10,
     "couples": 5,
     "validDraws": "440192",
     "pHatValid": 0.121305,
     "displayHatValid": "12.1%",
     "expectedHatTries": 8.243675,
     "expectedHatTriesDisplay": "8.24",
     "pNoSelfDrawBreaksCouples": 0.670259,
     "displayNoSelfDrawBreaksCouples": "67.0%",
     "validLoops": "112512",
     "pLoopValid": 0.310053,
     "displayLoopValid": "31.0%",
     "expectedShuffles": 3.225256,
     "expectedShufflesDisplay": "3.23"
    },
    {
     "people": 11,
     "couples": 0,
     "validDraws": "14684570",
     "pHatValid": 0.367879,
     "displayHatValid": "36.8%",
     "expectedHatTries": 2.718282,
     "expectedHatTriesDisplay": "2.72",
     "pNoSelfDrawBreaksCouples": 0,
     "displayNoSelfDrawBreaksCouples": "0%",
     "validLoops": "3628800",
     "pLoopValid": 1,
     "displayLoopValid": "100%",
     "expectedShuffles": 1,
     "expectedShufflesDisplay": "1.00"
    },
    {
     "people": 11,
     "couples": 1,
     "validDraws": "11881152",
     "pHatValid": 0.297648,
     "displayHatValid": "29.8%",
     "expectedHatTries": 3.359674,
     "expectedHatTriesDisplay": "3.36",
     "pNoSelfDrawBreaksCouples": 0.190909,
     "displayNoSelfDrawBreaksCouples": "19.1%",
     "validLoops": "2903040",
     "pLoopValid": 0.8,
     "displayLoopValid": "80.0%",
     "expectedShuffles": 1.25,
     "expectedShufflesDisplay": "1.25"
    },
    {
     "people": 11,
     "couples": 2,
     "validDraws": "9672904",
     "pHatValid": 0.242327,
     "displayHatValid": "24.2%",
     "expectedHatTries": 4.126661,
     "expectedHatTriesDisplay": "4.13",
     "pNoSelfDrawBreaksCouples": 0.341288,
     "displayNoSelfDrawBreaksCouples": "34.1%",
     "validLoops": "2338560",
     "pLoopValid": 0.644444,
     "displayLoopValid": "64.4%",
     "expectedShuffles": 1.551724,
     "expectedShufflesDisplay": "1.55"
    },
    {
     "people": 11,
     "couples": 3,
     "validDraws": "7917280",
     "pHatValid": 0.198345,
     "displayHatValid": "19.8%",
     "expectedHatTries": 5.041732,
     "expectedHatTriesDisplay": "5.04",
     "pNoSelfDrawBreaksCouples": 0.460844,
     "displayNoSelfDrawBreaksCouples": "46.1%",
     "validLoops": "1895040",
     "pLoopValid": 0.522222,
     "displayLoopValid": "52.2%",
     "expectedShuffles": 1.914894,
     "expectedShufflesDisplay": "1.91"
    },
    {
     "people": 11,
     "couples": 4,
     "validDraws": "6511008",
     "pHatValid": 0.163114,
     "displayHatValid": "16.3%",
     "expectedHatTries": 6.130664,
     "expectedHatTriesDisplay": "6.13",
     "pNoSelfDrawBreaksCouples": 0.556609,
     "displayNoSelfDrawBreaksCouples": "55.7%",
     "validLoops": "1543680",
     "pLoopValid": 0.425397,
     "displayLoopValid": "42.5%",
     "expectedShuffles": 2.350746,
     "expectedShufflesDisplay": "2.35"
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     "expectedHatTriesDisplay": "6.34",
     "pNoSelfDrawBreaksCouples": 0.571322,
     "displayNoSelfDrawBreaksCouples": "57.1%",
     "validLoops": "50989753530777600",
     "pLoopValid": 0.419168,
     "displayLoopValid": "41.9%",
     "expectedShuffles": 2.385677,
     "expectedShufflesDisplay": "2.39"
    },
    {
     "people": 20,
     "couples": 9,
     "validDraws": "346120851083310080",
     "pHatValid": 0.142267,
     "displayHatValid": "14.2%",
     "expectedHatTries": 7.029054,
     "expectedHatTriesDisplay": "7.03",
     "pNoSelfDrawBreaksCouples": 0.613279,
     "displayNoSelfDrawBreaksCouples": "61.3%",
     "validLoops": "45877376537395200",
     "pLoopValid": 0.377141,
     "displayLoopValid": "37.7%",
     "expectedShuffles": 2.651527,
     "expectedShufflesDisplay": "2.65"
    },
    {
     "people": 20,
     "couples": 10,
     "validDraws": "312426715251262464",
     "pHatValid": 0.128417,
     "displayHatValid": "12.8%",
     "expectedHatTries": 7.787113,
     "expectedHatTriesDisplay": "7.79",
     "pNoSelfDrawBreaksCouples": 0.650926,
     "displayNoSelfDrawBreaksCouples": "65.1%",
     "validLoops": "41303040523960320",
     "pLoopValid": 0.339537,
     "displayLoopValid": "34.0%",
     "expectedShuffles": 2.945185,
     "expectedShufflesDisplay": "2.95"
    }
   ],
   "headline": {
    "people": 10,
    "couples": 3,
    "validDraws": "672240",
    "pHatValid": 0.185251,
    "displayHatValid": "18.5%",
    "expectedHatTries": 5.398072,
    "expectedHatTriesDisplay": "5.40",
    "pNoSelfDrawBreaksCouples": 0.496435,
    "displayNoSelfDrawBreaksCouples": "49.6%",
    "validLoops": "175680",
    "pLoopValid": 0.484127,
    "displayLoopValid": "48.4%",
    "expectedShuffles": 2.065574,
    "expectedShufflesDisplay": "2.07"
   }
  },
  "lastYear": {
   "rule": "last year's draw was one loop (person 1 gave to 2, 2 to 3, ..., the last to 1); nobody may draw themselves or the person they had last year",
   "formulas": {
    "validDraws": "menage numbers: sum_j (-1)^j [2n/(2n-j)] C(2n-j, j) (n-j)!; P tends to e^-2 = 0.135335",
    "validLoops": "counted one loop at a time for 4-11 people"
   },
   "limit": 0.135335,
   "note": "Last year's draw is taken to be one loop (as the loop method gives). If last year's draw had mutual pairs, each such pair acts like a couple. Exact BigInt counts.",
   "rows": [
    {
     "people": 4,
     "validDraws": "2",
     "pHatValid": 0.083333,
     "displayHatValid": "8.33%",
     "expectedHatTries": 12,
     "expectedHatTriesDisplay": "12.00",
     "pNoSelfDrawRepeatsLastYear": 0.777778,
     "displayNoSelfDrawRepeatsLastYear": "77.8%",
     "validLoops": "1",
     "pLoopValid": 0.166667,
     "displayLoopValid": "16.7%",
     "expectedShuffles": 6,
     "expectedShufflesDisplay": "6.00"
    },
    {
     "people": 5,
     "validDraws": "13",
     "pHatValid": 0.108333,
     "displayHatValid": "10.8%",
     "expectedHatTries": 9.230769,
     "expectedHatTriesDisplay": "9.23",
     "pNoSelfDrawRepeatsLastYear": 0.704545,
     "displayNoSelfDrawRepeatsLastYear": "70.5%",
     "validLoops": "8",
     "pLoopValid": 0.333333,
     "displayLoopValid": "33.3%",
     "expectedShuffles": 3,
     "expectedShufflesDisplay": "3.00"
    },
    {
     "people": 6,
     "validDraws": "80",
     "pHatValid": 0.111111,
     "displayHatValid": "11.1%",
     "expectedHatTries": 9,
     "expectedHatTriesDisplay": "9.00",
     "pNoSelfDrawRepeatsLastYear": 0.698113,
     "displayNoSelfDrawRepeatsLastYear": "69.8%",
     "validLoops": "36",
     "pLoopValid": 0.3,
     "displayLoopValid": "30.0%",
     "expectedShuffles": 3.333333,
     "expectedShufflesDisplay": "3.33"
    },
    {
     "people": 7,
     "validDraws": "579",
     "pHatValid": 0.114881,
     "displayHatValid": "11.5%",
     "expectedHatTries": 8.704663,
     "expectedHatTriesDisplay": "8.70",
     "pNoSelfDrawRepeatsLastYear": 0.687702,
     "displayNoSelfDrawRepeatsLastYear": "68.8%",
     "validLoops": "229",
     "pLoopValid": 0.318056,
     "displayLoopValid": "31.8%",
     "expectedShuffles": 3.144105,
     "expectedShufflesDisplay": "3.14"
    },
    {
     "people": 8,
     "validDraws": "4738",
     "pHatValid": 0.11751,
     "displayHatValid": "11.8%",
     "expectedHatTries": 8.50992,
     "expectedHatTriesDisplay": "8.51",
     "pNoSelfDrawRepeatsLastYear": 0.680577,
     "displayNoSelfDrawRepeatsLastYear": "68.1%",
     "validLoops": "1625",
     "pLoopValid": 0.322421,
     "displayLoopValid": "32.2%",
     "expectedShuffles": 3.101538,
     "expectedShufflesDisplay": "3.10"
    },
    {
     "people": 9,
     "validDraws": "43387",
     "pHatValid": 0.119563,
     "displayHatValid": "12.0%",
     "expectedHatTries": 8.363796,
     "expectedHatTriesDisplay": "8.36",
     "pNoSelfDrawRepeatsLastYear": 0.674994,
     "displayNoSelfDrawRepeatsLastYear": "67.5%",
     "validLoops": "13208",
     "pLoopValid": 0.327579,
     "displayLoopValid": "32.8%",
     "expectedShuffles": 3.052695,
     "expectedShufflesDisplay": "3.05"
    },
    {
     "people": 10,
     "validDraws": "439792",
     "pHatValid": 0.121195,
     "displayHatValid": "12.1%",
     "expectedHatTries": 8.251173,
     "expectedHatTriesDisplay": "8.25",
     "pNoSelfDrawRepeatsLastYear": 0.670558,
     "displayNoSelfDrawRepeatsLastYear": "67.1%",
     "validLoops": "120288",
     "pLoopValid": 0.331481,
     "displayLoopValid": "33.1%",
     "expectedShuffles": 3.01676,
     "expectedShufflesDisplay": "3.02"
    },
    {
     "people": 11,
     "validDraws": "4890741",
     "pHatValid": 0.122523,
     "displayHatValid": "12.3%",
     "expectedHatTries": 8.161708,
     "expectedHatTriesDisplay": "8.16",
     "pNoSelfDrawRepeatsLastYear": 0.666947,
     "displayNoSelfDrawRepeatsLastYear": "66.7%",
     "validLoops": "1214673",
     "pLoopValid": 0.334731,
     "displayLoopValid": "33.5%",
     "expectedShuffles": 2.987471,
     "expectedShufflesDisplay": "2.99"
    },
    {
     "people": 12,
     "validDraws": "59216642",
     "pHatValid": 0.123625,
     "displayHatValid": "12.4%",
     "expectedHatTries": 8.088969,
     "expectedHatTriesDisplay": "8.09",
     "pNoSelfDrawRepeatsLastYear": 0.663952,
     "displayNoSelfDrawRepeatsLastYear": "66.4%"
    },
    {
     "people": 13,
     "validDraws": "775596313",
     "pHatValid": 0.124553,
     "displayHatValid": "12.5%",
     "expectedHatTries": 8.028688,
     "expectedHatTriesDisplay": "8.03",
     "pNoSelfDrawRepeatsLastYear": 0.661429,
     "displayNoSelfDrawRepeatsLastYear": "66.1%"
    },
    {
     "people": 14,
     "validDraws": "10927434464",
     "pHatValid": 0.125346,
     "displayHatValid": "12.5%",
     "expectedHatTries": 7.977929,
     "expectedHatTriesDisplay": "7.98",
     "pNoSelfDrawRepeatsLastYear": 0.659275,
     "displayNoSelfDrawRepeatsLastYear": "65.9%"
    },
    {
     "people": 15,
     "validDraws": "164806435783",
     "pHatValid": 0.12603,
     "displayHatValid": "12.6%",
     "expectedHatTries": 7.934607,
     "expectedHatTriesDisplay": "7.93",
     "pNoSelfDrawRepeatsLastYear": 0.657414,
     "displayNoSelfDrawRepeatsLastYear": "65.7%"
    },
    {
     "people": 16,
     "validDraws": "2649391469058",
     "pHatValid": 0.126627,
     "displayHatValid": "12.7%",
     "expectedHatTries": 7.897206,
     "expectedHatTriesDisplay": "7.90",
     "pNoSelfDrawRepeatsLastYear": 0.655792,
     "displayNoSelfDrawRepeatsLastYear": "65.6%"
    },
    {
     "people": 17,
     "validDraws": "45226435601207",
     "pHatValid": 0.127152,
     "displayHatValid": "12.7%",
     "expectedHatTries": 7.864591,
     "expectedHatTriesDisplay": "7.86",
     "pNoSelfDrawRepeatsLastYear": 0.654365,
     "displayNoSelfDrawRepeatsLastYear": "65.4%"
    },
    {
     "people": 18,
     "validDraws": "817056406224416",
     "pHatValid": 0.127618,
     "displayHatValid": "12.8%",
     "expectedHatTries": 7.835902,
     "expectedHatTriesDisplay": "7.84",
     "pNoSelfDrawRepeatsLastYear": 0.653099,
     "displayNoSelfDrawRepeatsLastYear": "65.3%"
    },
    {
     "people": 19,
     "validDraws": "15574618910994665",
     "pHatValid": 0.128033,
     "displayHatValid": "12.8%",
     "expectedHatTries": 7.81047,
     "expectedHatTriesDisplay": "7.81",
     "pNoSelfDrawRepeatsLastYear": 0.65197,
     "displayNoSelfDrawRepeatsLastYear": "65.2%"
    },
    {
     "people": 20,
     "validDraws": "312400218671253762",
     "pHatValid": 0.128406,
     "displayHatValid": "12.8%",
     "expectedHatTries": 7.787773,
     "expectedHatTriesDisplay": "7.79",
     "pNoSelfDrawRepeatsLastYear": 0.650955,
     "displayNoSelfDrawRepeatsLastYear": "65.1%"
    }
   ]
  },
  "helper": {
   "n": 10,
   "couples": 3,
   "singles": 4,
   "validDraws": "672240",
   "expectedShuffles": 5.398072,
   "expectedShufflesDisplay": "5.40",
   "pLastGivesToFirst": 0.116113,
   "lastGivesToFirstFraction": "9757/84030",
   "lastGivesToFirstLabel": "person 10 (single) gives to person 1 (has a partner)",
   "pLastGivesToFirstVsNoCouples": 0.111111,
   "fair": 0.111111,
   "fairFraction": "1/9",
   "pairTypes": [
    {
     "type": "single to single",
     "giver": 10,
     "receiver": 7,
     "pairs": 12,
     "p": 0.101107,
     "fraction": "1416/14005",
     "display": "10.1%",
     "ratioToOneOverNMinus1": 0.91
    },
    {
     "type": "single to someone with a partner",
     "giver": 10,
     "receiver": 1,
     "pairs": 24,
     "p": 0.116113,
     "fraction": "9757/84030",
     "display": "11.6%",
     "ratioToOneOverNMinus1": 1.045
    },
    {
     "type": "someone with a partner to a single",
     "giver": 1,
     "receiver": 10,
     "pairs": 24,
     "p": 0.116113,
     "fraction": "9757/84030",
     "display": "11.6%",
     "ratioToOneOverNMinus1": 1.045
    },
    {
     "type": "someone with a partner to a member of another couple",
     "giver": 1,
     "receiver": 3,
     "pairs": 24,
     "p": 0.133887,
     "fraction": "22501/168060",
     "display": "13.4%",
     "ratioToOneOverNMinus1": 1.205
    }
   ],
   "pMutualPair": 0.416637,
   "pMutualPairFraction": "1167/2801",
   "pMutualPairDisplay": "41.7%",
   "pMutualPairNoCouples": 0.393349,
   "note": "Exact, by enumerating every valid draw (10 people, couples 1&2, 3&4, 5&6, singles 7-10). Shuffling until the draw is valid makes every valid draw equally likely (rejection sampling), but with couples that is not the same as a uniformly random draw with nobody drawing themselves, and the giver-receiver chances are no longer all 1/(n-1). Each receiver is drawn by someone with total chance 1, spread over the givers allowed to draw them; a person with a partner has one allowed giver fewer, so each of those givers draws them a little more often. pairTypes: a single draws another single LESS often than 1/(n-1), a single and a partnered person draw each other a little more often, and a partnered person draws a member of another couple MORE often. Every pair of one type has the same chance (checked). pMutualPair = chance that at least two people draw each other in a valid draw, vs pMutualPairNoCouples with no couples."
  },
  "chart": {
   "note": "One row per group size: exact values. hatWorks = plain hat, nobody draws themselves; putBackDeadEnd = put-back rule leaves the last person with only their own name; mutualPair and singleLoop = in a uniformly random valid draw; couples3Hat = plain hat with 3 couples valid (n >= 6).",
   "byPeople": [
    {
     "people": 4,
     "hatWorks": 0.375,
     "hatFails": 0.625,
     "expectedTries": 2.666667,
     "putBackDeadEnd": 0.138889,
     "mutualPair": 0.333333,
     "singleLoop": 0.666667,
     "couples3Hat": null
    },
    {
     "people": 5,
     "hatWorks": 0.366667,
     "hatFails": 0.633333,
     "expectedTries": 2.727273,
     "putBackDeadEnd": 0.131944,
     "mutualPair": 0.454545,
     "singleLoop": 0.545455,
     "couples3Hat": null
    },
    {
     "people": 6,
     "hatWorks": 0.368056,
     "hatFails": 0.631944,
     "expectedTries": 2.716981,
     "putBackDeadEnd": 0.112778,
     "mutualPair": 0.396226,
     "singleLoop": 0.45283,
     "couples3Hat": 0.111111
    },
    {
     "people": 7,
     "hatWorks": 0.367857,
     "hatFails": 0.632143,
     "expectedTries": 2.718447,
     "putBackDeadEnd": 0.100532,
     "mutualPair": 0.385113,
     "singleLoop": 0.38835,
     "couples3Hat": 0.133333
    },
    {
     "people": 8,
     "hatWorks": 0.367882,
     "hatFails": 0.632118,
     "expectedTries": 2.718263,
     "putBackDeadEnd": 0.090495,
     "mutualPair": 0.394054,
     "singleLoop": 0.339783,
     "couples3Hat": 0.153175
    },
    {
     "people": 9,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718284,
     "putBackDeadEnd": 0.082382,
     "mutualPair": 0.394259,
     "singleLoop": 0.302032,
     "couples3Hat": 0.170282
    },
    {
     "people": 10,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.075646,
     "mutualPair": 0.393349,
     "singleLoop": 0.271828,
     "couples3Hat": 0.185251
    },
    {
     "people": 11,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.06996,
     "mutualPair": 0.393412,
     "singleLoop": 0.247117,
     "couples3Hat": 0.198345
    },
    {
     "people": 12,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.065093,
     "mutualPair": 0.393483,
     "singleLoop": 0.226523,
     "couples3Hat": 0.209835
    },
    {
     "people": 13,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.060876,
     "mutualPair": 0.393473,
     "singleLoop": 0.209099,
     "couples3Hat": 0.219963
    },
    {
     "people": 14,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.057186,
     "mutualPair": 0.393468,
     "singleLoop": 0.194163,
     "couples3Hat": 0.228935
    },
    {
     "people": 15,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.053929,
     "mutualPair": 0.393469,
     "singleLoop": 0.181219,
     "couples3Hat": 0.236926
    },
    {
     "people": 16,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.05103,
     "mutualPair": 0.393469,
     "singleLoop": 0.169893,
     "couples3Hat": 0.244078
    },
    {
     "people": 17,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.048434,
     "mutualPair": 0.393469,
     "singleLoop": 0.159899,
     "couples3Hat": 0.250512
    },
    {
     "people": 18,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.046095,
     "mutualPair": 0.393469,
     "singleLoop": 0.151016,
     "couples3Hat": 0.256326
    },
    {
     "people": 19,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.043976,
     "mutualPair": 0.393469,
     "singleLoop": 0.143067,
     "couples3Hat": 0.261604
    },
    {
     "people": 20,
     "hatWorks": 0.367879,
     "hatFails": 0.632121,
     "expectedTries": 2.718282,
     "putBackDeadEnd": 0.042047,
     "mutualPair": 0.393469,
     "singleLoop": 0.135914,
     "couples3Hat": 0.266414
    }
   ],
   "putBackLastRow10": [
    {
     "receiver": 1,
     "p": 0.091739,
     "display": "9.17%",
     "ratio": 0.8257
    },
    {
     "receiver": 2,
     "p": 0.09299,
     "display": "9.30%",
     "ratio": 0.8369
    },
    {
     "receiver": 3,
     "p": 0.094604,
     "display": "9.46%",
     "ratio": 0.8514
    },
    {
     "receiver": 4,
     "p": 0.096763,
     "display": "9.68%",
     "ratio": 0.8709
    },
    {
     "receiver": 5,
     "p": 0.099802,
     "display": "9.98%",
     "ratio": 0.8982
    },
    {
     "receiver": 6,
     "p": 0.104395,
     "display": "10.4%",
     "ratio": 0.9396
    },
    {
     "receiver": 7,
     "p": 0.112146,
     "display": "11.2%",
     "ratio": 1.0093
    },
    {
     "receiver": 8,
     "p": 0.128034,
     "display": "12.8%",
     "ratio": 1.1523
    },
    {
     "receiver": 9,
     "p": 0.179527,
     "display": "18.0%",
     "ratio": 1.6157
    }
   ]
  }
 },
 "checks": [
  {
   "name": "hat, 4 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.3746,
   "exact": 0.375,
   "se": 0.00108,
   "z": -0.37,
   "pass": true
  },
  {
   "name": "hat, 5 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.36677,
   "exact": 0.366667,
   "se": 0.00108,
   "z": 0.1,
   "pass": true
  },
  {
   "name": "hat, 6 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.36667,
   "exact": 0.368056,
   "se": 0.00108,
   "z": -1.28,
   "pass": true
  },
  {
   "name": "hat, 7 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.36729,
   "exact": 0.367857,
   "se": 0.00108,
   "z": -0.53,
   "pass": true
  },
  {
   "name": "hat, 8 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.368835,
   "exact": 0.367882,
   "se": 0.00108,
   "z": 0.88,
   "pass": true
  },
  {
   "name": "hat, 9 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.367215,
   "exact": 0.367879,
   "se": 0.00108,
   "z": -0.62,
   "pass": true
  },
  {
   "name": "hat, 10 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.367474,
   "exact": 0.367879,
   "se": 0.000482,
   "z": -0.84,
   "pass": true
  },
  {
   "name": "hat, 10 people: average number of people who draw their own name",
   "kind": "simulated vs exact",
   "simulated": 1.000406,
   "exact": 1,
   "se": 0.001,
   "z": 0.41,
   "pass": true
  },
  {
   "name": "hat, 10 people: exactly one person draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.368257,
   "exact": 0.367879,
   "se": 0.000482,
   "z": 0.78,
   "pass": true
  },
  {
   "name": "hat, 10 people with 3 couples: nobody draws themselves or their partner",
   "kind": "simulated vs exact",
   "simulated": 0.184711,
   "exact": 0.185251,
   "se": 0.000389,
   "z": -1.39,
   "pass": true
  },
  {
   "name": "hat, 10 people with 3 couples: nobody draws themselves but a couple draws each other",
   "kind": "simulated vs exact",
   "simulated": 0.182763,
   "exact": 0.182628,
   "se": 0.000386,
   "z": 0.35,
   "pass": true
  },
  {
   "name": "hat, 10 people: nobody draws themselves or last year's person (last year one loop)",
   "kind": "simulated vs exact",
   "simulated": 0.121117,
   "exact": 0.121195,
   "se": 0.000326,
   "z": -0.24,
   "pass": true
  },
  {
   "name": "hat, 11 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.366895,
   "exact": 0.367879,
   "se": 0.00108,
   "z": -0.91,
   "pass": true
  },
  {
   "name": "hat, 12 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.36477,
   "exact": 0.367879,
   "se": 0.00108,
   "z": -2.88,
   "pass": true
  },
  {
   "name": "hat, 13 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.366195,
   "exact": 0.367879,
   "se": 0.00108,
   "z": -1.56,
   "pass": true
  },
  {
   "name": "hat, 14 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.36754,
   "exact": 0.367879,
   "se": 0.00108,
   "z": -0.31,
   "pass": true
  },
  {
   "name": "hat, 15 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.37037,
   "exact": 0.367879,
   "se": 0.00108,
   "z": 2.31,
   "pass": true
  },
  {
   "name": "hat, 16 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.36895,
   "exact": 0.367879,
   "se": 0.00108,
   "z": 0.99,
   "pass": true
  },
  {
   "name": "hat, 17 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.36792,
   "exact": 0.367879,
   "se": 0.00108,
   "z": 0.04,
   "pass": true
  },
  {
   "name": "hat, 18 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.367095,
   "exact": 0.367879,
   "se": 0.00108,
   "z": -0.73,
   "pass": true
  },
  {
   "name": "hat, 19 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.369835,
   "exact": 0.367879,
   "se": 0.00108,
   "z": 1.81,
   "pass": true
  },
  {
   "name": "hat, 20 people: nobody draws their own name",
   "kind": "simulated vs exact",
   "simulated": 0.367865,
   "exact": 0.367879,
   "se": 0.00108,
   "z": -0.01,
   "pass": true
  },
  {
   "name": "redraw rule, 10 people: average number of full draws until nobody has themselves",
   "kind": "simulated vs exact",
   "simulated": 2.7189,
   "exact": 2.718282,
   "se": 0.00558,
   "z": 0.11,
   "pass": true
  },
  {
   "name": "redraw rule, 10 people: needs more than 3 full draws",
   "kind": "simulated vs exact",
   "simulated": 0.251647,
   "exact": 0.25258,
   "se": 0.00112,
   "z": -0.83,
   "pass": true
  },
  {
   "name": "redraw rule, 10 people: person 1 gives to person 10",
   "kind": "simulated vs exact",
   "simulated": 0.111127,
   "exact": 0.111111,
   "se": 0.000811,
   "z": 0.02,
   "pass": true
  },
  {
   "name": "redraw rule, 10 people: person 1 equally likely to give to each of the other 9 (chi-square, 8 df)",
   "kind": "uniformity (chi-square)",
   "simulated": 2.16444,
   "exact": 8,
   "se": 4,
   "z": -1.95,
   "pass": true,
   "zMethod": "Wilson–Hilferty"
  },
  {
   "name": "redraw rule, 10 people: at least one pair draws each other",
   "kind": "simulated vs exact",
   "simulated": 0.392913,
   "exact": 0.393349,
   "se": 0.00126,
   "z": -0.35,
   "pass": true
  },
  {
   "name": "redraw rule, 10 people: one single loop through everyone",
   "kind": "simulated vs exact",
   "simulated": 0.271467,
   "exact": 0.271828,
   "se": 0.00115,
   "z": -0.31,
   "pass": true
  },
  {
   "name": "redraw rule, 10 people: average number of separate loops",
   "kind": "simulated vs exact",
   "simulated": 2.024807,
   "exact": 2.023832,
   "se": 0.00206,
   "z": 0.47,
   "pass": true
  },
  {
   "name": "redraw rule, 5 people: all 44 valid draws equally likely (chi-square, 43 df)",
   "kind": "uniformity (chi-square)",
   "simulated": 42.80956,
   "exact": 43,
   "se": 9.27,
   "z": 0.05,
   "pass": true,
   "zMethod": "Wilson–Hilferty"
  },
  {
   "name": "put-back rule, 4 people: the last person is left with only their own name",
   "kind": "simulated vs exact",
   "simulated": 0.137755,
   "exact": 0.138889,
   "se": 0.000773,
   "z": -1.47,
   "pass": true
  },
  {
   "name": "put-back rule, 5 people: the last person is left with only their own name",
   "kind": "simulated vs exact",
   "simulated": 0.13279,
   "exact": 0.131944,
   "se": 0.000757,
   "z": 1.12,
   "pass": true
  },
  {
   "name": "put-back rule, 6 people: the last person is left with only their own name",
   "kind": "simulated vs exact",
   "simulated": 0.112375,
   "exact": 0.112778,
   "se": 0.000707,
   "z": -0.57,
   "pass": true
  },
  {
   "name": "put-back rule, 8 people: the last person is left with only their own name",
   "kind": "simulated vs exact",
   "simulated": 0.08957,
   "exact": 0.090495,
   "se": 0.000642,
   "z": -1.44,
   "pass": true
  },
  {
   "name": "put-back rule, 10 people: the last person is left with only their own name",
   "kind": "simulated vs exact",
   "simulated": 0.075542,
   "exact": 0.075646,
   "se": 0.000264,
   "z": -0.39,
   "pass": true
  },
  {
   "name": "put-back rule, 10 people: names put back per attempt",
   "kind": "simulated vs exact",
   "simulated": 1.087971,
   "exact": 1.09038,
   "se": 0.00128,
   "z": -1.89,
   "pass": true
  },
  {
   "name": "put-back rule, 10 people: the last to draw gets the second-to-last",
   "kind": "simulated vs exact",
   "simulated": 0.179327,
   "exact": 0.179527,
   "se": 0.000399,
   "z": -0.5,
   "pass": true
  },
  {
   "name": "put-back rule, 10 people: the last to draw gets the first",
   "kind": "simulated vs exact",
   "simulated": 0.09192,
   "exact": 0.091739,
   "se": 0.0003,
   "z": 0.6,
   "pass": true
  },
  {
   "name": "put-back rule, 10 people: the first to draw gets the last",
   "kind": "simulated vs exact",
   "simulated": 0.119823,
   "exact": 0.120204,
   "se": 0.000338,
   "z": -1.13,
   "pass": true
  },
  {
   "name": "put-back rule, 10 people: whom the last to draw gets, against the exact (unequal) chances (chi-square, 8 df)",
   "kind": "goodness of fit (chi-square)",
   "simulated": 8.703963,
   "exact": 8,
   "se": 4,
   "z": 0.34,
   "pass": true,
   "zMethod": "Wilson–Hilferty"
  },
  {
   "name": "put-back rule, 10 people: the last two to draw get each other",
   "kind": "simulated vs exact",
   "simulated": 0.015684,
   "exact": 0.015854,
   "se": 0.00013,
   "z": -1.31,
   "pass": true
  },
  {
   "name": "put-back rule, 10 people: at least one pair draws each other",
   "kind": "simulated vs exact",
   "simulated": 0.392158,
   "exact": 0.39161,
   "se": 0.000508,
   "z": 1.08,
   "pass": true
  },
  {
   "name": "put-back rule, 10 people: one single loop through everyone",
   "kind": "simulated vs exact",
   "simulated": 0.27202,
   "exact": 0.272628,
   "se": 0.000463,
   "z": -1.31,
   "pass": true
  },
  {
   "name": "put-back rule, 12 people: the last person is left with only their own name",
   "kind": "simulated vs exact",
   "simulated": 0.064305,
   "exact": 0.065093,
   "se": 0.000552,
   "z": -1.43,
   "pass": true
  },
  {
   "name": "put-back rule, 12 people: the last to draw gets the second-to-last",
   "kind": "simulated vs exact",
   "simulated": 0.149766,
   "exact": 0.150427,
   "se": 0.000826,
   "z": -0.8,
   "pass": true
  },
  {
   "name": "put-back rule, 12 people: the last to draw gets the first",
   "kind": "simulated vs exact",
   "simulated": 0.076489,
   "exact": 0.076428,
   "se": 0.000614,
   "z": 0.1,
   "pass": true
  },
  {
   "name": "put-back rule, 12 people: the first to draw gets the last",
   "kind": "simulated vs exact",
   "simulated": 0.098665,
   "exact": 0.097239,
   "se": 0.000685,
   "z": 2.08,
   "pass": true
  },
  {
   "name": "put-back rule, 12 people: the last two to draw get each other",
   "kind": "simulated vs exact",
   "simulated": 0.010554,
   "exact": 0.011177,
   "se": 0.000243,
   "z": -2.56,
   "pass": true
  },
  {
   "name": "put-back rule, 12 people: whom the last to draw gets, against the exact (unequal) chances (chi-square, 10 df)",
   "kind": "goodness of fit (chi-square)",
   "simulated": 2.534038,
   "exact": 10,
   "se": 4.47,
   "z": -2.31,
   "pass": true,
   "zMethod": "Wilson–Hilferty"
  },
  {
   "name": "put-back rule, 15 people: the last person is left with only their own name",
   "kind": "simulated vs exact",
   "simulated": 0.05361,
   "exact": 0.053929,
   "se": 0.000505,
   "z": -0.63,
   "pass": true
  },
  {
   "name": "put-back rule, 15 people: the last to draw gets the second-to-last",
   "kind": "simulated vs exact",
   "simulated": 0.122011,
   "exact": 0.1213,
   "se": 0.00075,
   "z": 0.95,
   "pass": true
  },
  {
   "name": "put-back rule, 15 people: the last to draw gets the first",
   "kind": "simulated vs exact",
   "simulated": 0.061259,
   "exact": 0.06132,
   "se": 0.000551,
   "z": -0.11,
   "pass": true
  },
  {
   "name": "put-back rule, 15 people: the first to draw gets the last",
   "kind": "simulated vs exact",
   "simulated": 0.076205,
   "exact": 0.0755,
   "se": 0.000607,
   "z": 1.16,
   "pass": true
  },
  {
   "name": "put-back rule, 15 people: the last two to draw get each other",
   "kind": "simulated vs exact",
   "simulated": 0.007407,
   "exact": 0.007294,
   "se": 0.000196,
   "z": 0.58,
   "pass": true
  },
  {
   "name": "put-back rule, 15 people: whom the last to draw gets, against the exact (unequal) chances (chi-square, 13 df)",
   "kind": "goodness of fit (chi-square)",
   "simulated": 12.441021,
   "exact": 13,
   "se": 5.1,
   "z": 0.02,
   "pass": true,
   "zMethod": "Wilson–Hilferty"
  },
  {
   "name": "put-back rule, 20 people: the last person is left with only their own name",
   "kind": "simulated vs exact",
   "simulated": 0.041895,
   "exact": 0.042047,
   "se": 0.000449,
   "z": -0.34,
   "pass": true
  },
  {
   "name": "put-back rule, 20 people: the last to draw gets the second-to-last",
   "kind": "simulated vs exact",
   "simulated": 0.092547,
   "exact": 0.091993,
   "se": 0.00066,
   "z": 0.84,
   "pass": true
  },
  {
   "name": "put-back rule, 20 people: the last to draw gets the first",
   "kind": "simulated vs exact",
   "simulated": 0.046613,
   "exact": 0.046308,
   "se": 0.00048,
   "z": 0.63,
   "pass": true
  },
  {
   "name": "put-back rule, 20 people: the first to draw gets the last",
   "kind": "simulated vs exact",
   "simulated": 0.0549,
   "exact": 0.054942,
   "se": 0.000521,
   "z": -0.08,
   "pass": true
  },
  {
   "name": "put-back rule, 20 people: the last two to draw get each other",
   "kind": "simulated vs exact",
   "simulated": 0.003924,
   "exact": 0.004209,
   "se": 0.000148,
   "z": -1.92,
   "pass": true
  },
  {
   "name": "put-back rule, 20 people: whom the last to draw gets, against the exact (unequal) chances (chi-square, 18 df)",
   "kind": "goodness of fit (chi-square)",
   "simulated": 17.912022,
   "exact": 18,
   "se": 6,
   "z": 0.1,
   "pass": true,
   "zMethod": "Wilson–Hilferty"
  },
  {
   "name": "put-back rule, 5 people: all 44 completed draws against their exact (unequal) chances (chi-square, 43 df)",
   "kind": "goodness of fit (chi-square)",
   "simulated": 28.283958,
   "exact": 43,
   "se": 9.27,
   "z": -1.74,
   "pass": true,
   "zMethod": "Wilson–Hilferty"
  },
  {
   "name": "loop, 10 people: share of draws that are one single loop",
   "kind": "sanity",
   "simulated": 1,
   "exact": 1,
   "se": 0,
   "z": 0,
   "pass": true
  },
  {
   "name": "loop, 10 people: share of draws in which someone has themselves",
   "kind": "sanity",
   "simulated": 0,
   "exact": 0,
   "se": 0,
   "z": 0,
   "pass": true
  },
  {
   "name": "loop, 10 people: share of draws with a pair drawing each other",
   "kind": "sanity",
   "simulated": 0,
   "exact": 0,
   "se": 0,
   "z": 0,
   "pass": true
  },
  {
   "name": "loop, 10 people: person 1 gives to person 10",
   "kind": "simulated vs exact",
   "simulated": 0.111608,
   "exact": 0.111111,
   "se": 0.000314,
   "z": 1.58,
   "pass": true
  },
  {
   "name": "loop, 10 people: person 1 equally likely to give to each of the other 9 (chi-square, 8 df)",
   "kind": "uniformity (chi-square)",
   "simulated": 8.637668,
   "exact": 8,
   "se": 4,
   "z": 0.32,
   "pass": true,
   "zMethod": "Wilson–Hilferty"
  },
  {
   "name": "loop, 10 people with 3 couples: no couple next to each other in the loop",
   "kind": "simulated vs exact",
   "simulated": 0.483925,
   "exact": 0.484127,
   "se": 0.0005,
   "z": -0.4,
   "pass": true
  },
  {
   "name": "loop, 10 people: nobody gets last year's person (last year one loop)",
   "kind": "simulated vs exact",
   "simulated": 0.33016,
   "exact": 0.331481,
   "se": 0.000471,
   "z": -2.81,
   "pass": true
  },
  {
   "name": "loop, 5 people: all 24 possible loops equally likely (chi-square, 23 df)",
   "kind": "uniformity (chi-square)",
   "simulated": 17.7094,
   "exact": 23,
   "se": 6.78,
   "z": -0.75,
   "pass": true,
   "zMethod": "Wilson–Hilferty"
  },
  {
   "name": "helper, 10 people with 3 couples: average shuffles until a valid draw",
   "kind": "simulated vs exact",
   "simulated": 5.38952,
   "exact": 5.398072,
   "se": 0.0154,
   "z": -0.56,
   "pass": true
  },
  {
   "name": "helper, 10 people with 3 couples: person 10 gives to person 7 (single to single)",
   "kind": "simulated vs exact",
   "simulated": 0.10082,
   "exact": 0.101107,
   "se": 0.000953,
   "z": -0.3,
   "pass": true
  },
  {
   "name": "helper, 10 people with 3 couples: person 10 gives to person 1 (single to someone with a partner)",
   "kind": "simulated vs exact",
   "simulated": 0.11595,
   "exact": 0.116113,
   "se": 0.00101,
   "z": -0.16,
   "pass": true
  },
  {
   "name": "helper, 10 people with 3 couples: person 1 gives to person 10 (someone with a partner to a single)",
   "kind": "simulated vs exact",
   "simulated": 0.11499,
   "exact": 0.116113,
   "se": 0.00101,
   "z": -1.11,
   "pass": true
  },
  {
   "name": "helper, 10 people with 3 couples: person 1 gives to person 3 (someone with a partner to a member of another couple)",
   "kind": "simulated vs exact",
   "simulated": 0.13427,
   "exact": 0.133887,
   "se": 0.00108,
   "z": 0.36,
   "pass": true
  },
  {
   "name": "helper, 10 people with 3 couples: at least one pair draws each other",
   "kind": "simulated vs exact",
   "simulated": 0.41769,
   "exact": 0.416637,
   "se": 0.00156,
   "z": 0.68,
   "pass": true
  },
  {
   "name": "helper, 6 people with 2 couples: all 116 valid draws equally likely (chi-square, 115 df)",
   "kind": "uniformity (chi-square)",
   "simulated": 143.46288,
   "exact": 115,
   "se": 15.2,
   "z": 1.78,
   "pass": true,
   "zMethod": "Wilson–Hilferty"
  }
 ]
}